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arXiv:1904.02590v1 [math.DG] 4 Apr 2019 apnsi weak in mappings M¨uller and h ope tutr iegst h onayo h ouispa moduli the valu of functional boundary Willmore the to the diverges that structure is complex observation the approach mainly alternate Their an [17][26]. gave see they theorem, compactness the With h as curvature Gauss the φ uhthat such value opcns hnmnnotie n[2.I eea,tettlc total the general, In [22]. in Willmor obtained the phenomenon minimizing compactness surfaces of existence the got he topology, fixed .iie[5 rvdtecmatestermo mesn maps immersing of theorem compactness the proved [25] T.Rivire sa lotfltLpcizgahotieo oesaltplgcldisk topological small functional some Willmore of the and outside to volume equal graph bounded is Lipschitz with curvature flat total a almost says an which is 2.1] lemma [27, theorem bu the about C hoe n[] epoe hta immersion an d that proved high He a considered [2]. compa Breuning in the Recently, theorem deduced and convergence. spaces graphical tangent of the of radius) uniform with where ooe Sobolev fa mesn umnfl.I h rtclcase critical the In . immersing an of k ( n > p nte motn bevto fimrigsrae ihfiieto finite with surfaces immersing of observation important Another n1] agrfis rvdsrae with surfaces proved first Langer In[18], Let RMVHUDRFCNEGNETERMO UFCSIN SURFACES OF THEOREM CONVERGENCE GROMOV-HAUSDORFF A Σ : k A F → Abstract. n rv o-olpigvrino H´elein’s convergenc of th version decomposition non-collapsing a Simon’s prove Leon and explain can we applications, opcns hoe fsc ufcsi intrinsic in esti surfaces the such Using of theorem radius. isothermal compactness call a we which quantity new in sa is ) stescn udmna omand form fundamental second the is Σ : v L R Σ vra siae h ad omof norm Hardy the Sver´ak estimated ˇ n ∈ p g → ( ihlcluiombuddae n ml oa curvature total small and area bounded uniform local with p ovre,te ol regard could they converges, L C R ∞ ≥ 1 n sn this Using . ,α )nr ftescn udmna form. fundamental second the of norm 2) W nti ae,w anysuytecmatesadlclstru local and compactness the study mainly we paper, this In ea meso ftesraeΣin Σ surface the of immersion an be gahi nfr ml alfor ball small uniform a in -graph W loc R 2 Σ , 2 2 ,p (Σ | K → \ | dµ S, C IHSALTTLCURVATURE TOTAL SMALL WITH R g 1 L u ne h condition the under But . , n 1 ∞ − topology. ) etmto ftemti,EKwr,R c¨tl,YX i[6[4 an [16][14] Li Sch¨atzle, Y.X. R. E.Kuwert, metric, the of -estimation n p fw eadtescn udmna oma h scn derivat “second the as form fundamental second the regard we if INI U,JEZHOU JIE SUN, JIANXIN 1. f k k A −△ ill W Z g Introduction ◦ k Z f = φ L Σ | A v p : k ∗ | dF ( M | A = ≤ ob ofra n osdrtecnegneo these of convergence the consider and conformal be to Kdµ f 2 R 1 | ) Σ ≤ 2 L n ∗ C ⊗ p dµ < p Kdµ | → ( H tplg n extrinsic and -topology 4 dF < α g = > p 8 πε, g R | n ovdteequation the solved and theorem[12][14]. e π. , R 2 n n g dµ steidcdmti.Teeaemn results many are There metric. induced the is n h oa uvtr of curvature total The . )aelocally are 2) aeo h stemlrdu eestablish we radius isothermal the of mate oe[7 ntevepito convergence of viewpoint the in eorem[27] 1 + ,Lo io rvdadecomposition a proved Simon Leon 2, = g l − pt osatfracoe ufc with surface closed a for constant a to up ihbuddvlm and volume bounded with n ce p R feitneo h iloeminimizer, Willmore the of existence of Σ hsi emti nlgeo the of analogue geometric a is This . f raueol otosthe controls only urvature teso uhsrae ntemeaning the in surfaces such of ctness mninlgnrlzto fLanger’s of generalization imensional ∩ M Σ : B 1 g (0) a uvtr stecompensated the is curvature tal g n netecmlxstructure complex the once And . functional. e tr fimrigsurfaces immersing of cture → | .Uigti n oiigthe noticing and this Using s. ucetsalttlcurvature total small sufficient C A iljm vrtegp8 gap the over jump will e | W 1 R 2 gah vrsalballs(but small over -graphs e nrdeti a is ingredient key A . n 2 , 2 ihWlmr functional Willmore with wa oooy As topology. -weak F sdfie by defined is R M L | 1 A nr of -norm | p dµ R π n ive” g as ≤ d 2 J. SUN, J. ZHOU

Geometrically, the divergency of the complex structure means the collapsing of geodesics, which changes the local topology(or area density) and contributes a gap to the Willmore functional value. To use a geometric quantity to replace the convergence of the complex structure to rule out the collapsing phenomenon, we define the isothermal radius: Definition 1.1 (Isothermal radius). Assume (Σ,g) is a Riemannian surface with metric g. For p Σ, define the isothermal radius of the metric g at the point p to be ∀ ∈ i (p) = sup r isothermal coordinate f : D (0) U(p) s.t. U(p) BΣ(p) and f (0) = p , g { |∃ 0 1 → ⊃ r 0 } Σ where Br (p) is the geodesic ball centered at p with radius r. We call ig(Σ) = infp Σ ig(p) the isothermal radius of (Σ,g). ∈ By the existence theorem of local conformal coordinates, i(p) > 0 for any p Σ, but it may depend on the (Σ,g) and the point p. If we assume the isothermal radius of∈ a sequence of metrics has uniform lower bound, we can estimate the uniform bound of the metric and deduce the following compactness theorem: Theorem 1.2 (Fundamental convergence theorem). For any two positive real numbers R, V and ε < 1, and some positive function i0 : (0, R] R>0, define (n,ε,i0,V,R) as the space of Riemannian → C n immersions (of open Riemannian surfaces) F : (Σ,g,p) R proper in BR(0) such that F (p)=0, 1 → F − (BR(0)) is connected, 1 i (x) i (R F (x) ) > 0, x Σ F − (B (0)), g ≥ 0 − | | ∀ ∈ ∩ R A 2 4πε, −1 Σ F (BR(0)) | | ≤ Z ∩ and 1 2 µg(Σ F − (BR(0))) V R , n ∩ ≤ where BR(0) is the open ball in R with radius R and ig is the isothermal radius. Then for any sequence n Fk : (Σk,gk,pk) (R , 0) k∞=1 in (n,ε,i0,V,R), there exists a subsequence(also denoted as Σk) such that{ → } C 1) (Intrinsic convergence) There is a pointed Riemannian surface (Σ,g,p) (may not complete) with con- 1 p tinuous Riemannian metric g such that (Σk Fk− (BR(pk)),gk,pk) converges in pointed L topology ∩ p to (Σ,g,p), that is, there exist smooth Φ :Σ Σ , s.t. Φ∗g g in L (dµ ). k → k k k → loc g 2) The complex structure k of Σk converges locally to the complex structure of Σ. 3) (Extrinsic convergence)OThere exists a proper isometric immersion F : (ΣO,g,p) B (0) such that → R F (p)=0 and F Φ converges to F weakly in W 2,2(Σ, Rn) and strongly in W 1,p(Σ, Rn). k ◦ k loc loc Moreover, just as Anderson noticed in [1, main Lemma 2.2], Theorem 1.2 can feed back to provide a lower bound estimate of the isothermal radius.

Proposition 1.3. For any fixed V R+, there exist ε0(V ) > 0 and α0(V ) > 0 such that for any properly immersed Riemannian surface F : (Σ∈ ,g,p) (Rn, 0) satisfying → 2 µg(Σ B1(0)) V and A dµg ε0(V ), ∩ ≤ Σ B (0) | | ≤ Z ∩ 1 we have ig(x) α0(V ), x Σ B 1 (0), ≥ ∀ ∈ ∩ 2 n where B1(0) = y R y < 1 and by Σ B1(0) we actually mean the connected component of 1 { ∈ || | } ∩ Σ F − (B (0)) containing p. ∩ 1 As an application, we use the fundamental convergence theorem to give a blowup approach for Leon Simon’s decomposition theorem. A key point is to use a blowup argument to reduce the immersing case to the embedding case. Another key point is to use the Poincar´einequality to estimate the area of the surface out of the multi-graph, which finally guarantees the multi-graph must be single, see Theorem 5.2 for more details. A GROMOV-HAUSDORFF CONVERGENCE THEOREM OF SURFACES IN Rn WITH SMALL TOTAL CURVATURE 3

As another application of the estimation of the isothermal radius, we can prove a non-collapsing version of H´elein’s compactness theorem. In [12], H´elein proved that a sequence of W 2,2-conformal immersions 2,2 with uniform bounded Willmore energy will converge weakly to a limit f in Wloc (Σ S) outside of finite many singularities S Σ. But the conformal invariant property of the∞ Willmore energy\ and the non-compactness of both the⊂ intrinsic and extrinsic conformal groups (Σ) and (Rn) may cause the 2 M 2 M limit f to collapse. In the case Σ = S , if we assume additionally volgk (S ) 1, then we can use the estimation∞ of isothermal radius to exclude the collapsing caused by the extrinsic≡ conformal group (Rn), see Corollary 6.2. This result is also obtained in [6] by the bubble tree convergence argument. MThe article is organized as following. In sect.2, we prepared some important preliminaries. In sect.3, we use the lower bound of the isothermal radius to estimate the metric. In sect.4, we prove the fundamental convergence theorem. In sect.5 and sect.6, we prove Proposition 1.3 and give the two applications.

2. Preliminaries and Notations 2.1. Hardy estimate. In this section, we list some theorems we will quote in this paper. The first is M¨uller and Sver´ak’sˇ Hardy-estimate [22, Corollary 3.5.7.] mentioned above. Under a local isothermal coordinate, we assume g = e2u(dx2 + dy2), and the Gauss curvature equation is

u = Kdµ = G∗ω, −△ ∗ g ∗ n 1 e1(p)+√ 1e2(p) where G is the Gauss map G :Σ CP − , G(p) = [ 2− ], e1,e2 are orthonormal basis of Σ at → n 1 { } the point p and ω is the K¨aller form on CP − . In [22], when observing that ω has the algebraic structure 2n 1 2n 1 n 1 of determinant when transgressed to the total space S − of the Hopf fibration π : S − CP − , 1 1 → M¨uller and Sver´akˇ improved the regularity of Kdµg from L to , the Hardy space, by using the results of Coifman, Meyer, Lions and Semmes[3](see∗ also [23]) underH the condition

A 2 4πε | | ≤ Z for some ε (0, 1). Moreover, when noticing that in dimension two, the fundamental solution ln x ∈ 2u | | belongs to BMO, the dual of Hardy space ([9]), they can solve v = Ke with L∞-estimation: −△ 1,2 n Theorem 2.1. For 0 <ε< 1. Assume ϕ W0 (C, CP ) satisfies C ϕ∗ω =0 and that C Dϕ Dϕ ∈ 2 1 | ∧ |≤ 1 2 4n (1 ε n ) 2πε. Then ϕ∗ω (C) with ϕ∗ω 1 c1C(n,ε) Dϕ L2 , whereRC(n,ε)=1+ (1 −εR)2 . Moreover, ∗ ∈ H k kH ≤ k k − the equation u = ϕ∗ω admits a unique solution v : C R which is continuous and satisfies: −△ ∗ → lim v(z)=0, z →∞ and 2 1 2 2 2 D v + Dv + max v (z) c2 ϕ∗ω 1 c3C(n,ε) Dϕ L2 , C | | { C | | } z C | | ≤ k kH ≤ k k Z Z ∈ where c1,c2 and c3 are constants independent of n and ε, which will be denoted as a same notation c in the following text. 2.2. Monotonicity formulae. We will also quote Leon Simon’s monotonicity formulae [27](see also [15]) in the blowup argument of estimating the isothermal radius. It is also used in our application II. Theorem 2.2. Assume µ = 0 is an integral 2-varifold in an open set U Rn with square integrable 6 2 n ⊂ weak mean curvature Hµ L (µ) and Bρ0 (x0) U for some x0 R and ρ0 > 0. Then for δ > 0 and 0 < σ ρ ρ , we have∈ ⊂⊂ ∈ ∀ ≤ ≤ 0 2 2 1 σ− µ(B (x )) (1 + δ)ρ− µ(B (x )) + (1 + )W (µ), σ 0 ≤ ρ 0 4δ where W (µ)= 1 H 2dµ is the Willmore energy of µ. 4 U | µ| Here we explainR some notations we use in this paper. For a Riemannian immersion F : (Σ,g) Rn, p Σ, x Rn and R> 0, we denote → ∈ ∈ B (x)= y Rn y x < R , BΣ(p)= q Σ d (q,p) < R , R { ∈ || − | } R { ∈ | g } 4 J. SUN, J. ZHOU

and R 1 Σ (p) := the connected component of F − (BR(F (p))) containing p. We will also abuse the inaccurate but intuitionistic notation Σ B (F (p)) to denote ΣR(p). ∩ R

3. L∞–estimate of the metric under isothermal coordinates 3.1. Basic setting: for an immersing Riemannian surface F : Σ Rn with the induced metric g = dF dF , by the existence theorem of isothermal coordinate [7] and→ Riemann mapping theorem, there exists⊗ an isothermal coordinate f : D (0) U Σ Rn for a small neighborhood of p such that 0 1 → ⊆ → f0(0) = p, where D1 = z = x + √ 1y C : z < 1 is the unit disk on the complex plain and { − ∈ | | } n the scale of U may depend on (Σ,g,p). This means f0 : D1 R is a conformal immersion, i.e., 2u(z) 2 2 2u → 2u f0∗g = e (dx + dy ). Furthermore, we have f0∗dµg = e dx dy, G∗ω = Kdµg = Ke dx dy and 2 1 2 n 1 ∧ n 1 ∧ DG g = 2 A g, where G : Σ CP − is the Gauss map, ω is the K¨ahler form on CP − and A is the | | | | → n second fundamental form of the immersion F :Σ R . As a consequence, if we let ϕ = G f0, then we get the basic Gauss curvature equation: → ◦ 2u u = Ke = f ∗(Kdµ )= f ∗G∗ω = ϕ∗ω −△ ∗ 0 g ∗ 0 ∗ . 2 Lemma 3.1. If A < + and Jϕ dx dy = Dϕ Dϕ πε for some ε (0, 1), then f0(D1) g D1 D1 2u | | ∞ | | ∧ 1 | ∧ |≤ ∈ Ke = ϕ∗ω could be extended to be a function w (C) with ∗ R R ∈ H R 2 w 1(C) c(n,ε) A g. k kH ≤ | | Zf0(D1) 1 Furthermore, the equation u = w admits a unique solution v = − w which is continuous on C and satisfies: −△ −△ lim v(z)=0, z →∞ and 2 1 2 2 2 D v + Dv + max v (z) c w 1 c(n,ε) A g. C | | { C | | } z C | | ≤ k kH ≤ | | Z Z ∈ Zf0(D1) n 1 n 1 Proof. Set ϕ = G f : D Σ CP − , and defineϕ ¯ : C CP − by ◦ 0 1 → → → ϕ(z),z D1, ϕ¯(z)= 1 ∈ ϕ( ),z D∗ = z z 1 . ( z¯ ∈ 1 { || |≥ } Then we have,

ϕ¯∗ω = ϕ∗ω ϕ∗ω =0, C − Z ZD1 ZD1 Dϕ¯ Dϕ¯ dxdy =2 Dϕ¯ Dϕ¯ dxdy 2πε, C | ∧ | | ∧ | ≤ Z ZD1 and 2 2 2 Dϕ¯ dxdy =2 Dϕ dxdy A gdµg. C | | | | ≤ | | Z ZD1 Zf0(D1) The work is done if we define w := ϕ¯∗ω and apply Theorem 2.1.  ∗ 2 Remark 3.2. If A dµg 4πε, then f0(D1) | | ≤ R 1 2 Dϕ Dϕ dxdy = det(Dϕ)∗Dϕdxdy A dµ πε. | ∧ | ≤ 4 | | g ≤ ZD1 ZD1 q Zf0(D1) 2 2 So, in the following text, we may use A dµg 4πε to replace the conditions A < + f0(D1) | | ≤ f0(D1) | |g ∞ and Jϕ dx dy = Dϕ Dϕ πε. D1 | | ∧ D1 | ∧ |≤R R TheR following corollaryR is simple but important in the estimation of the isothermal radius. A GROMOV-HAUSDORFF CONVERGENCE THEOREM OF SURFACES IN Rn WITH SMALL TOTAL CURVATURE 5

2 Corollary 3.3. In the case Ak dµg 0, vk converges to 0 uniformly on the unit D1. Σk | |gk k → The following lemma showsR that the lower bound of isothermal radius could rule out the collapsing phenomenon and dominate the metric uniformly.

2u(z) 2 Lemma 3.4. Take a local isothermal coordinate f0 : (D1, 0) (U Σ,p) and denote f0∗g = e (dx + 2 2 → ⊂ dy ) as before. Assume A dµg A0, Dϕ Dϕ dxdy πε, areag(f0(D1)) V and f0(D1) | | ≤ D1 | ∧ | ≤ ≤ d (p,f (D )) i > 0. Then u L (D ) and for any r (0, 1), there exists a C = C(n,ε,i ,V,A , r) g 0 1 0 R loc∞ 1 R 0 0 such that ≥ ∈ ∈

sup u C(n,ε,i0,V,A0, r). Dr | |≤ Proof. Let v = w and h = u v as above. Then h is harmonic and it is equal to estimate u and h since v has been−△ estimated by Lemma− 3.1. Step 1. Estimate the upper bound. x D1, let r(x)= d(x,∂D1)=1 x > 0. Then Jensen’s inequality and mean value theorem for harmonic∀ ∈ functions imply −| | u(x)=¯u(x) v¯(x)+ v(x) − 1 u(y)+2C(n,ε)A ≤ πr2(x) 0 ZDr(x) (x) 1 ln e2u(y) +2C(n,ε)A ≤ πr2(x) 0 ZDr(x) (x) ln V +2C(n,ε)A0 =: C(n,ε,V,A0, r) ≤ π(1 r)2 − ¯ 1 for x Dr, where h(x)= 2 h(y) andu ¯,v ¯ are defined similarly. ∈ πr (x) Dr(x) (x) Step 2. The argument developed in [5] gives a control ofu ¯(0) from below by d (p,f (D )), i.e.,u ¯(0) C R g 0 1 ≥− for some constant C = C(n,ε,i0,V,A0) > 0. We argue by contradiction. If there exist a sequence of Riemannian immersions F : (Σ ,g ,p ) Rn which admit isothermal coordinates f : (D , 0) k k k k → k 1 → (Uk,pk) and satisfy 2 Ak dµgk A0, Dϕk Dϕk dxdy πε, Σ f (D ) | | ≤ D | ∧ | ≤ Z k ∩ k 1 Z 1

area(Σk fk(D1)) V, dgk (pk,∂Uk) i0, gk ∩ ≤ ≥

1 2uk(z) 2 2 butu ¯k(0) = uk = Ck , where uk is defined by f ∗gk = e (dx + dy ). Then, Lemma 3.1 π D1 → −∞ k and step 2. imply R h (0) = h¯ (0) =u ¯ (0) v¯ (0) u¯ (0) + C(n,ε)A k k k − k ≤ k 0 → −∞ and h (x)= u (x) v (x) C(n,ε,A ,V,r) for x D (0). k k − k ≤ 0 ∈ r Now, Harnack’s inequality and Lemma 3.1 again imply u = h + v ⇒ on D , which means k k k −∞ r lim areagk (Dr)=0 , for any r (0, 1). But on the other hand, for δ<δ0(n,ε,A0,V,i0) small enough, k ∈ d→∞(p ,∂U ) i implies that for x ∂D (0), gk k k ≥ 0 ∀ ∈ 1 i d (x, 0) 0 ≤ gk l (γ ) (γ (t)= tx : [0, 1] D ) ≤ gk 0,x 0,x → 1 1 = euk(tx) x dt | | Z0 δ 1 δ 1 − = euk(tx)dt + euk(tx)dt + euk(tx)dt 0 δ 1 δ − Z Z Z 1 1 2 2 ln V 1 δ 1 δ +2C(n,ε)A0 − − 1 δe 2π(1−δ)2 + e2uk (tx)tdt dt ≤ δ δ t  Z   Z  6 J. SUN, J. ZHOU

1 1 1 2 1 1 2 + e2uk (tx)tdt dt 1 δ 1 δ t Z − Z −    1  1 1 δ 2 i0 − 1 δ 2 + e2uk(tx)tdt ln − ≤ 2 δ δ  Z 1    1 2 1 2 + e2uk (tx)tdt ln (1 δ) , 1 δ − −  Z −    which further implies 1 δ 1 i0 2 − 1 δ 2 e2uk(tx)tdt ln − +2 e2uk(tx)tdt ln (1 δ) . 2 ≤ δ δ 1 δ − − Z Z −        Integrating this on θ = x [0, 2π], we get ∈ 2 2π 1 δ 2πi − 0 2 e2uk (tx)tdt ln (1 δ) ln δ 4 ≤ − − Z0  Zδ 1    + e2uk(tx)tdt ln (1 δ) dθ 1 δ − −  Z −   2areag (D1 δ Dδ)( ln δ)+2V ( ln (1 δ)). ≤ k − \ − − − Take δ0 small again and we get 2 πi0 areag (D1 δ Dδ) > 0 k − \ ≥−8 ln δ

for δ<δ0(n,ε,A0,V,i0). This contradicts to lim areagk (Dr) = 0. k Step 3. Finally, Lemma 3.1, Step 2 and Step 3→∞ imply

max h(x) C(n,ε,A0,V,r) and h(0) > C(n,ε,A0,V,i0, r). x Dr ≤ − ∈ Using Harnack’s inequality and Lemma 3.1 again, we know u(x) C(n,ε,A ,V,i , r), x D . ≥− 0 0 ∀ ∈ r 

The following corollary follows immediately since h is harmonic.

Corollary 3.5. Under the above setting, h k C(n,ε,A ,V,i , r, k) for any integer k. k kC (Dr ) ≤ 0 0 4. Intrinsic and extrinsic convergence In this section, we first use Gromov’s compactness theorem and the estimations established in the last section to prove an intrinsic convergence theorem and then use the W 2,2-estimate for the mean curvature equation to prove an extrinsic convergence theorem. They together form the fundamental convergence Theorem 1.2.

4.1. Intrinsic convergence. Definition 4.1. (n,ε,i , A , V )= F : (Σ,g,p) Rn properly immersing i (Σ) i > 0, E 0 0 { → | g ≥ 0 Dϕ Dϕ πε, A 2 A , area (U(p)) V, p Σ | ∧ |≤ | | ≤ 0 g ≤ ∀ ∈ } ZD1 ZU(p) where ϕ = G f0 is the Gauss map in the coordinate defined as in the beginning of last section and f : D (0) U◦(p) is defined as in the definition of the isothermal radius. 0 1 → We conclude the lemmas in the last section as below. A GROMOV-HAUSDORFF CONVERGENCE THEOREM OF SURFACES IN Rn WITH SMALL TOTAL CURVATURE 7

Lemma 4.2. For any (F : (Σ,g,p) Rn) (n,ε,i , A , V ) and r (0, 1) and r (0, 1), there exists → ∈E 0 0 ∈ ∈ a constant C = C(n,ε,i0, A0,V,r) s.t.

u ∞ + Du 2 C(n,ε,i , A ,V,r), k kL (Dr) k kL (Dr ) ≤ 0 0 2u where u is defined by f ∗g = e g and f : D (0) U(p) is the isothermal coordinate defined above. 0 0 0 1 → The following theorem is a modification of the Fundamental theorem in Petersen’s book[24](See also [4]) in a weaker regularity condition.

Theorem 4.3 (Compactness theorem for immersed Riemannian surfaces). Any pointed sequence Fk : n { (Σk,gk,pk) R k∞=1 in (n,ε,i0, A0, V ) admits a subsequence(still denoted as (Σk,gk,pk) k∞=1) which converges to→ a complete} RiemannianE surface (Σ,g,d,p) with a continuous Riemannian{ metric} g and a compatible metric d d in the following sense ∼ g (a) (Σk,gk,pk) converges to (Σ,d,p) in pointed Gromov-Hausdorff topology; (b) (Σk,pk) converges to (Σ,p) as pointed Riemannian surface(i.e., the complex structure (hence the smooth structure) converges); p (c) p (1, + ), (Σk,gk,pk) converges to (Σ,g,p) in pointed-L (hence a.e.) topology. ∀ ∈ ∞ 2 (d) In general, we have d dg Cd. But in the case Ak dµg 0, we have (Σk,gk,pk) converges ≤ ≤ Σk | | k → to (Σ,g,p) in C0 topology and d = d. loc g R Before proving this theorem, we make some conceptions clear.

Definition 4.4 (Convergence of complex(differential) structure). Assume (Σk,pk) k∞=1 and (Σ,p) are m+1,β m+1{ ,β } all pointed Riemannian(C∞, C ) surfaces admitting complex(C∞, C ) structures

∞ = f : D (0) U Σ ∞ , U =Σ , Ok { ks r → ks ⊂ k}s=1 ks k s=1 [ ∞ = f : D (0) U Σ ∞ , U =Σ O { s r → s ⊂ }s=1 s s=1 [ m+1,β respectively. (WLOG, we assume pk Uk1 and p U1.) We call the complex (C∞, C ) structure ω ∈ m+1,β ∈ 1 k converges to in pointed C (C∞, C ) topology if K Dom(fs− ft), k0 large enough, Ok k , O ∀ ⊂⊂ ◦ ∃ ∀ ≥ 0 ω ∞ m+1,β 1 1 Cc (Cc ,Cc ) 1 K Dom(f − f ) and f − f f − f . ⊂⊂ ks ◦ kt ks ◦ kt −−−−−−−−−−−→ s ◦ t In this case, we also call the sequence (Σ ,p ) ∞ converges to (Σ,p) as pointed Riemannian(C∞, { k k }k=1 Cm+1,β) surface.

ω Remark 4.5. Assume k and are complex structures. Then by Montel’s theorem, k in C topology in CO0 topology(afterO passing to a subsequence). O → O ⇔ Ok → O p Definition 4.6 (L convergence). 1) Assume A is a compact set in a differential manifold M, fk k∞=1 p { } and f are functions on M. We call fk converges to f in L (A) (a.e.) if there exists a finite coordinate cover x : U U˜ Rn N with N U A s.t. { s s → s ⊂ }s=1 ∪s=1 s ⊃ p ˜ 1 Lloc(US ) 1 f˜ = f x− f˜ = f x− , 1 s N. ks k ◦ s −−−−−−→ s ◦ s ∀ ≤ ≤ It is easy to check this definition does not depend on the choice of coordinates. Lp(A) 2) For Riemannian metrics gk and g on M, we say gk g if for the above coordinates xs, the 1 s α−−−−→β 1 s α β coefficients of the tensors defined by xs− ∗gk = gkαβdx dx and xs− ∗g = gαβdx dx satisfies

s s g g p 0, 1 s N, K U˜ . k kαβ − αβkL (K) → ∀ ≤ ≤ ∀ ⊂⊂ s 1 α,β n ≤X≤ 8 J. SUN, J. ZHOU

3) Assume (Mk,gk,pk) and (M,g,p) are complete Riemannian with continuous Riemannian pointed Lp metrics. We say (M ,g ,p ) − (M,g,p) if R > 0, Ω BM (p) M and embedding k k k −−−−−−−−→ ∀ ∃ ⊃ R ⊂ F :Ω M for k large enough s.t. BMk (p ) F (Ω) M and k → k R k ⊂ k ⊂ k Lp(Ω) F ∗g g. k k −−−−→ Proof of Theorem 4.3. Step 1. (Gromov-Haustorff convergence) In the first paragraph we omit the footprint k. Let f0 : D1(0) U(p) be an isothermal coordinate 2u ∞ → and define u by f0∗g = e g0. Then, we have u L (Dr ) C(r) = C(n,ε,i0, A0,V,r) by Lemma 4.2, hence for any curve γ : [0, 1] D with γ(0) =kx,k γ(1) = y≤, we have → r C(r) C(r) e− l (γ) l (f (γ)) e l (γ), g0 ≤ g 0 ≤ g0 which means (a) d(f (x),f (y)) eC(r) x y , x, y D (0), 0 0 ≤ | − | ∀ ∈ r C(r) (b) d(f (x),f (y)) e− min x y , 2r x y , x, y D (0). 0 0 ≥ {| − | − | | − | |} ∀ ∈ r The same argument as in [24, sec 10.3.4] implies the capacity estimate, i.e., R > 0 and ((Σ,g,p) n 1 C( r+1 ) ∀ ∀ → R ) , N(α) = N(α, R, r, C(r)) and δ = e− 2 r s.t. Cap (α) N(α) for α α = δ, ∈ E ∃ 10 BR(p) ≤ ∀ ≤ 0 where the capacity CapBR(p)(α) is defined by α Cap (α) = maximum number of disjoint balls in X X 2 − n for compact metric space X. So, for a sequence (Σk,gk,pk) R k∞=1 , choosing conformal k { → } k ⊂k E Σk k coordinates covering fks : D1(0) Uks(p ) Σk ∞ s.t. fks(0) = p , p = pk, Uks B (p ), { → s ⊂ }k,s=1 s 1 ⊃ i0 s N l Σk = sfks(Dr(0)) and B δ (pk) fks(Dr(0)), Gromov’s compactness theorem[11](see also [24, sec l 2 s ∪ · ⊂ ∪ ¯ 10.1.4] then guarantees the sequence (Bl δ (p ),gk,pk) k∞=1 converges(after passing to a subsequence) { · 2 k } to a metric space (B¯ δ (p), dl,p) in pointed Gromov-Haustorff topology. W.L.O.G., one could assume l 2 ¯ ¯ · (Bl δ (p), dl,p) (Bl+1 δ (p), dl+1,p). After taking direct limit, we have · 2 ⊂ · 2 p GH (Σk,gk,pk) = lim B¯ δ (pk) − lim B¯ δ (p)=:(X,d,p). l 2 l 2 −→ · −−−−→ −→ · From now on, we assume all (Σk,gk,pk) and (Σ,d,p) are in a same metric space Y locally since Gromov-Haustorff convergence is equal to Haustorff convergence after passing to a subsequence. Step 2.(Convergence of complex structure) k 2uks = Assume fks : D1(0) Uks(ps )(֒ Y ) are the isothermal coordinates taken above and e g0 →∂f 2 ∂f →2 ∞ ks ks ∞ uks L (Dr ) fks∗ gk = dfks, dfks = ( ∂x + ∂y )g0. Then, Lemma 4.2 implies fks L (Dr ) ek k C(r) h i | | | | k∇ k ≤ ≤ e , i.e., fks k∞=1 are all local Lipschitz with uniform Lipschitz constant. So Arzela-Ascoli’s lemma { } C0,β (D ) implies (after passing to a subsequence) f r f : D (0) Y . Furthermore, we have ks −−−−−−→ s r → C(r) d(fs(x),fs(y)) = lim d(fks(x),fks(y)) e x y , k ≤ | − | →∞ and C(r) C(r) r σ d(f (x),f (y)) e− min x y , 2r x y e− − x y , s s ≥ {| − | − | | − | |} ≥ σ | − | C(r) for x, y Dσ(0) Dr(0), i.e., fs is local bilipschitz with LipDr fs e and hence also injective. If we ∈ ⊂⊂ ≤ k define ps = limk pks, then when noticing that all Xk are length spaces, one know Uks(ps ) converges →∞ 0 X C to some Us(ps) B (ps) X in Haustorff topology as subsets in Y . So fks fs : D¯ r(0) Y implies ⊃ i0 ⊂ −−→ → Im(fs) X, i.e., fs : (D1(0), 0) (Us(ps)) X is an injective map from a to a Haustorff space, hence⊂ is an embedding. → ⊂ A GROMOV-HAUSDORFF CONVERGENCE THEOREM OF SURFACES IN Rn WITH SMALL TOTAL CURVATURE 9

Moreover, we claim f : D (0) U (p ) is surjective (hence homeomorphic). In fact, for x U (p )= s 1 → s s ∀ ∈ s s r limk fks(Dr) (Haustorff convergence as subsets in Y ), r (0, 1) and xk =: fks(ak) fks(Dr(0)) s.t.∪ x →∞x. W.L.O.G., we can assume a a D¯ (0). Then∃ ∈ ∈ k → k → ∈ r 0 d(f (a), x) d(f (a ),f (a)) + d(f (a ), x )+ d(x , x) ≤ s ≤ s k s s k k k eC(r) a a + d(f (a ),f (a )) + d(x , x) 0, ≤ | k − | s k ks k k → since fks fs uniformly on compact subsets of D1(0). So, x = fs(a) and fs is surjective. This means Σ := X is→ a topological manifold. 1 To construct a complex structure on X, we consider the transport function fks− fkt with domain 1 1 ◦ 1 1 Dom(fks− fkt) Dom(fs− ft) in Haustorff topology as subsets in C. We have d(fks− fkt(z),fs− ◦ 1 → 1 ◦ 1 1 ◦ ◦ ft(z)) d(fks− fkt(z),fks− ft(z)) + d(fks− ft(z),fs− ft(z)) 0 uniformly on compact subsets of ≤ ◦ 0 ◦ ◦ ◦ → 1 Cc 1 Dom(f − f ) since f f and f − are locally uniformly bilipschitz. This means s ◦ t kt −−→ t ks 0 1 Cc 1 1 f − f f − f on Dom(f − f ). ks ◦ kt −−→ s ◦ t s ◦ t 1 But we know fks− fkt is analytic since fks and fkt are both conformal coordinates in their intersection ◦ 1 domain, so Montel’s theorem implies f − f is also analytic on its domain. That means = f : s ◦ t O { s D1(0) s∞=1 is a complex structure on Σ and } ω (Σ , ,p ) C (Σ, ,p), k Ok k −−→ O k as pointed Riemannian surface, where k = fks : Dr(0) Uks(ps ) Σk s∞=1. Step 3.(Riemannian metric on Σ) O { → ⊂ } n For the isothermal coordinate fks : D1 Uks Σk R with pull back metric represented as 2uks 2uks → ⊂ → 1 fks∗ gk = e g0. Recall uks = Kke = wks in D1 for some wks (C). As before, we define 1 −△ n ∈ H v = − w and h = u v . Then, (Σ R ) and Lemma 3.1 implies ks −△ ks ks ks − ks k → ∈E wks 1(C) C(n,ε,A0,i0). k kH ≤ Furthermore, by Lemma 3.1, we have

vks L∞(C) + vks W 1,2(C) C wks 1(C) C(n,ε,A0,i0). k k k k ≤ k kH ≤ 1 Now, Rellich’s lemma and the weak compactness of (C)(see [28, chap 3.5.1]) imply there exist ws 1(C) and v W 1,2(C) s.t. H ∈ H s ∈ Lp(hence a.e.) w ⇀ w as distribution, and v v , p (1, + ). ks s ks −−−−−−−−−→ s ∀ ∈ ∞ Moreover, for any ϕ C∞(C), we have ∈ c

vs ϕ = lim vks ϕ = lim wksϕ = wsϕ, C ∇ ∇ k C ∇ ∇ k C C Z →∞ Z →∞ Z Z 1,2 i.e., vs W0 (C) satisfies the weak equation vs = ws in C. And weak lower semi-continuity of the ∈ 1 −△ norm of the Banach space = (V MO)∗ imply ws 1(C) lim inf k wks 1(C) C(n,ε,A0,i0). H k kH ≤ →∞ k kH ≤ So, by Lemma 3.1 again, we get v C0(C) and s ∈ vs L∞(C) + vs W 1,2(C) C ws 1(C) C(n,ε,A0,i0). H k k k k ≤ k k ≤ ∞ Cc (D1) On the other hand, by Corollary 3.5, there exists hs harmonic on D1 such that hks hs. Denote p −−−−−→ Lloc(D1) 0 ∞ us := hs +vs. Then we get uks us and us C (D1) with us L (Dr ) C(n,ε,A0,i0,V,r), r (0, 1). −−−−−−→ ∈ k k ≤ ∀ ∈ With this us, we can construct a continuous local metric g on Σ by defining 1 ∗ 2us g = (fs− ) (e g0), where fs : D1 Us(ps) is a coordinate in the complex structure constructed in Step 2. In fact, Step 2. → ∞ O Cc 1 1 1 ∗ 2us 1 ∗ 2ut also claim k , i.e., fks− fkt fs− ft, from which we can get (fs− ) (e g0) = (ft− ) (e g0) a.e.(henceO everywhere→ O since both◦ are−−→ continuous)◦ on their common domain, i.e., the metric g is globally well defined. Moreover, uks us a.e. on D1 and us L∞(D ) C(n,ε,A0,i0,V,r) imply limk lg (γ)= → k k r ≤ →∞ k 10 J. SUN, J. ZHOU

1 lg(γ), γ C ([0, 1], Σ) by dominate convergence theorem. Thus the induced metric dg is compatible with the∀ limit∈ metric d. More precisely, we have

dg(x, y) = inf lg(γ) = inf lim lgk (γ) lim dgk (x, y)= d(x, y), x, y Σ, γ γ k ≥ ∀ ∈ →∞ where we use the Gromov-Haustorff convergence result from Step 1. in the last equation. On the other ∞ ∞ hand, since uks L (Dr ) + us L (Dr ) C(n,ε,A0,i0,V,r), one know all the metrics are uniformly equivalent tok thek Euclideank metrick in every≤ coordinate and d Cd g ≤ for some C = C(n,ε,A0,i0, V ). 2 0 Moreover, in the case Ak dµg 0, by Corollary 3.3, we know uks converges to us in C (D1).So Σk | | k → loc for any δ > 0, u(x) δ u (x) u(x)+δ for large k independent on x D (0). For any γ : [0, 1] D (0), − ≤ Rk ≤ ∈ r → r 1 1 1 δ u γ u γ δ u γ e− e ◦ γ′ e k◦ γ′ e e ◦ γ′ . | |≤ | |≤ | | Z0 Z0 Z0 Since the last estimate is uniform for all γ, we can take infimum for γ joining x and y and then let k and δ 0 to get → ∞ → dg(x, y) = lim dgk (x, y)= d(x, y). k →∞ Step 4.(Lp-convergence of the metric structure) 1 By gluing the local diffeomorphisms φks = fks fs− : Us D1 Uks which converge to identity together by partition of unity, we get the following global◦ descriptio→ n of→ pointed-convergence of differential structure.

Lemma 4.7. If (Σk, k,pk) converges to (Σ, ,p) as pointed differential surfaces, then for any fixed l, there exist differentialO maps Φ : Ω = l UO Ω = l U for k large enough, such that Φ are kl l ∪s=1 s → kl ∪s=1 ks kl embeddings when restricted to compact subsets of Ωl. Proof. (It can be found in [24, sec 10.3.4], we write it here for the convenience of readers.) Assume 1 k = fks : D1 Uks s∞=1 and = fs : D1 Us s∞=1. Define φks : fks fs− : Us D1 Uks. OThen for{ t = s, if→U }U = ∅, whenO { putting φ→ : U} U in local coordinates◦ f : →D →U and 6 s ∩ t 6 kt t → kt s 1 → s fks : D1 Uks of Σ and Σk, we have → ∞ 1 1 1 Cc 1 1 φ˜ = f − φ f = f − f f − f f − f f − f = id. kt ks ◦ kt ◦ s ks ◦ kt ◦ t ◦ s −−→ s ◦ t ◦ t ◦ s That is, the local map φkt between Σ and Σk converges smoothly to identity w.r.t. the differential ˆ 1 structures and k. That is, if we denote φks = fks− φks, then for any compact subset K Dom(φ ) ODom(φO )= U U and integer m, ◦ ⊂ ks ∩ kt s ∩ t φˆ φˆ m φˆ id m + id φˆ m 0, as k + . k kt − kskC (K) ≤k kt − kC (K) k − kskC (K) → → ∞ Now, choose a partition of unity λ , λ for U ,U , i.e., smooth functions λ , λ on Σ with suppλ { 1 2} { s t} 1 2 1 ⊂ Us, suppλ2 Ut and λ1 + λ2 = 1 on Us Ut. Then λ1 = 1 on Us Ut and λ2 = 1 on Ut Us. Let ˆ ˆ ⊂ ˆ ˆ ˆ ∪ ˆ ˆ ˆ \ ˆ \ Φk = λ1φks + λ2φkt. Then Φk φks = (λ1 1)φks + λ2φkt = λ2(φkt φks) 0 in Cc∞(Us). For the ˆ ˆ − − ˆ − →ˆ same reason, Φk φkt 0 in Cc∞(Ut). But we know φks id in Cc∞(Us) and φkt id in Cc∞(Ut), so if we define − → → → fks φˆks on Us, Φk = ◦ f Φˆ on U U , ( kt ◦ k t\ s then Φk is well defined and for any compact subset K Us Ut, Φk : K Φk(K) Uks Ukt is a diffeomorphism for k large enough. ⊂⊂ ∪ → ⊂ ∪ The above argument show that we can glue two sequences of diffeomorphisms together if they are arbitrary close for k large enough. So, by induction, for the finite sequences of local coordinates φ l : { ks}s=1 Us Uks, they are all close to the identity(hence to each other) for k large enough. When denoting → l l Ωl = s=1Us and Ωkl = s=1Uks, we can glue them together to be a local diffeomorphism Φkl :Ωl Ωkl ∪ 1 ∪ → such that Φ− φ converges smoothly to the identity on its domain.  kl ◦ ks A GROMOV-HAUSDORFF CONVERGENCE THEOREM OF SURFACES IN Rn WITH SMALL TOTAL CURVATURE11

p p Lloc L By this lemma, we have (Φ )∗g g since (φ )∗g g by the construction of g and Φ is kl k −−−→ ks k −−→ kl arbitrary close to φks in Us for k large enough. That is, (Σ ,g ,p ) (Σ,g,p) (X,d,p) in pointed Lp topology. k k k → ∼ − 2 In the case Ak dµg 0, by gluing the local uniform convergence of uks us, we know (Σk,gk,pk) Σk | | k → → converges to (Σ,g,p) in C0 topology.  R loc n + 4.2. Extrinsic convergence. Assume Fk : (Σk,gk,pk) R k=1∞ converges to (Σ,g,p) in intrinsic p { → } p pointed-L -topology. Then there exist Φ : (Σ,p) (Σ ,p ) such that Φ∗g g in L . Moreover, one k → k k k k → loc could use Xk := Fk Φk(they have common domain Σ) to represent the immersion Fk and ask whether ◦ k,p n these Xk converge in W (Σ, R ). In this sense, we have the following extrinsic convergence theorem. n + Theorem 4.8. Assume Fk : (Σk,gk,pk) R k=1∞ (n,ε,A0,i0, V ) is the sequence converges to (Σ,g,p) in pointed-Lp-topology{ as in theorem→ 4.3} with⊂F E (p ) 0. Then there exists a isometric k k ≡ immersion F :Σ Rn such that F converges to F weakly in W 2,2(Σ, Rn) and strongly in W 1,p(Σ, Rn). → k loc loc Proof. Step 1.(local convergence) Take notations as in the proof of the intrinsic convergence: i0 = the common k isothermal radius, fks : D1 Uks(ps ) Σk and fs : D1 Us(ps) Σ local isothermal coordinates 2uks → ⊂ 2us → ⊂ 1 k with metrics fks∗ gk = e g0 and fs∗g = e g0 respectively, φks = fks fs− : Us(ps) Uks(ps ) p ◦ → L n diffeomorphism such thatg ˜k = φks∗ gk g. Assume Fk : (Σk,pk1) R to be the immersion of Σk n −−→ → k n into R such that Fk(pk1) = 0 and regard Fks = Fk φks : Us(ps) Uks(ps ) R as immersion(not necessarily isometry) of U (p ) into Rn. Then under◦ the isothermal→ coordinate→f : D U (p ), if we s s s 1 → s s define F˜ = F f : D Rn, then ks ks ◦ s 1 → F˜ = e2uks F˜ = e2uks H~ =: h , △ ks △gk ks k ks where we use the mean curvature equation in the last equation. Note that

h 2 = e4uks H~ 2 e2C(n,ε,A0,V,r) A 2 eCA | ks| | ks| ≤ | k| ≤ 0 ZDr ZDr ZΣk and F˜ (x) F˜ (x) F˜ (0) dF˜ x eCr. | k1| ≤ | k1 − k1 | ≤ | k1|| |≤ By local L2-estimation for elliptic equation, we get ˜ ˜ F 1 2,2 ′ C(r, r′)( F 1 2 + h 2 ) C(r) < . k k kW (Dr ) ≤ k k kL (Dr ) k kskL (Dr ) ≤ ∞ ˜ 2,2 n ˜ ˜ 2,2 n Thus, there exists an F 1 Wloc (D1, R ) such that Fk1 converges to F 1 weakly in Wloc (D1, R ) and 1,p n∞ ∈ ˜ ˜ p ∞ strongly in Wloc (D1, R ). Hence Fk∗1gRn converges to F ∗ 1gRn in Lloc(D1). But we know ∞ p F˜∗ gRn = (F φ f )∗gRn = f ∗ g f ∗g in L (D ). k1 k ◦ k1 ◦ 1 k1 k → 1 loc 1 ˜ ˜ 1 n So, F ∗ 1gRn = f1∗g, i.e., F 1 := F 1 f1− : U1(p1) R is a local Riemannian immersion. ∞ ∞ ∞ ◦ → Step 2. (extending the local limit to global) We argue by induction. Let Ω1 = U1(p1), Ωl+1 =

Ωl ps Ωl Us(ps); Φk1 = φk1, Φkl : Ωl Ωkl Σk constructed in Lemma 4.7. Now, assume we have ∪ ∪ ∈ → ⊂ n constructed isomorphism immersion F l :Ωl R such that Xkl := Fk Φkl converges to F l weakly 2,2 n 1,p ∞ n → β n ◦ ∞ in W (Ωl, R ) and strongly in W (Ωl, R ) Cc (Ωl, R ). Then for any ps Ωl, Fk(ps) F l(ps). loc loc ∩ ∈ → ∞ Noticing F˜ F˜ (p ) + eC r C(r) in D and repeating the progress in Step 1, we know F = F | ks| ≤ | ks s | ≤ r ks k ◦ φ ( F Φ ) converges weakly in W 2,2(U (p ), Rn) and strongly in W 1,p(U , Rn) Cβ (U (p ), Rn). ks ≈ k ◦ k(l+1) loc s s loc s ∩ c s s Moreover, by construction of Φkl, we have 1 1 X− F =Φ− φ id in C∞(Ω B (p )). kl ◦ ks kl ◦ ks → c l ∩ i0 s Thus Fks and Xkl converges to the same limit in their common domain, i.e., Xk(l+1) := Fk Φk(l+1) 2,2 n 1,p n β ◦ n converges to some F l+1 weakly in Wloc (Ωl+1, R ) and strongly in Wloc (Ωl+1, R ) Cc (Ωl+1, R ), and n ∞ ∩ F l+1 : Ωl+1 R is an isometric immersion which extends F l. The work is done by taking a direct ∞ → ∞ limit F = limF l.  −→ ∞ 12 J. SUN, J. ZHOU

Corollary 4.9. By Langer’s weak lower semi-continuous theorem[18], we also know

2 2 A lim inf Ak A0 | | ≤ k | | ≤ ZΣ →∞ ZΣk under the above assumption. Proof. The result can be found in [18], [21], we write the proof here for the convenience of the reader. The key observation is, as a function of (ξ,ζ,η) = (F,DF,D2F ), J(ξ,ζ,η) := A 2dµ depends on | g|g g η = D2F convexly for each fixed ξ and ζ, since it is an nonnegative quadratic form of D2F in any fixed coordinate. For any compact domain D Σ, we suppose A 2dµ < . Since F converges to ⊂⊂ D | |g g ∞ k F weakly in W 2,2(Σ, Rn) and strongly in W 1,p(Σ, Rn), by Lusin’s theorem, Egorov’s theorem and the loc loc R absolute continuity of integral, there exists a compact set S D such that F,DF,D2F are continuous ⊂⊂ on S, (Fk, DFk) converges to (F,DF ) uniformly on S and

A 2dµ A 2dµ ε. | |g g ≥ | |g g − ZS ZD 2 2 In the case D A gdµg = , we can take S A gdµg M for any M > 0. By the convexity of J we know | | ∞ | | ≥ R R J(F , DF ,D2F ) J(F , DF ,D2F )+ D J(F,DF,D2F ) (D2F D2F ) k k k ≥ k k η · k − + [D J(F , DF ,D2F ) D J(F,DF,D2F )] (D2F D2F ). η k k − η · k − 2 2 Since F,DF,D F C(S) and dF dF is a metric, we know DηJ(F,DF,D F ) C(S). By the weak 2 ∈ 2 2 ⊗ ∈ L convergence of D Fk to D F , we get

D J(F,DF,D2F ) (D2F D2F ) 0. η · k − → ZS 2 Moreover, the uniform convergence of (Fk, DFk) to (F,DF ) on S and uniform boundness of D Fk and 2 2 2 2 D F in L imply that DηJ(Fk, DFk,D F ) converges to DηJ(F,DF,D F ) uniformly on S and

[D J(F , DF ,D2F ) D J(F,DF,D2F )] (D2F D2F ) 0. η k k − η · k − → ZS As a result, we get

2 2 lim inf J(Fk, DFk,D Fk) lim inf J(Fk, DFk,D F ) k ≥ k →∞ ZS →∞ ZS = J(F,DF,D2F ) ZS J(F,DF,D2F ) ε. ≥ − ZD Letting ε 0 and D Σ, we get → → 2 2 A lim inf Ak . | | ≤ k | | ZΣ →∞ ZΣk 

4.3. Proof of Theorem 1.2. In this subsection, we will add the above two subsections to prove Theorem 1.2. The key work we need to do in the local case is to use the bounded volume condition to control the limit mapping such that it is proper.

r 1 Proof. Recall we use Σ (p ) to denote the connected component of F − (B (0)) Σ containing p and k k k r ∩ k k use BΣk (p ) to denote the open geodesic ball in Σ centered at p with radius r. Then for each r (0, R], r k k k ∈ BΣk (p ) Σr (p ). Since i (x) i (R F (x) ), by the same argument as in the global case(the above r k ⊂ k k gk ≥ 0 − | k | two subsections), we know there exists a (non-complete) pointed Riemannian surface (Σr,g,d,p) with n continuous metric g such that dg d and an isometric immersion Fr : (Σr,p) (R , 0) such that, after passing to a subsequence, ∼ → A GROMOV-HAUSDORFF CONVERGENCE THEOREM OF SURFACES IN Rn WITH SMALL TOTAL CURVATURE13

Σk p 1) (Br (pk),gk,pk) converges to (Σr,g,p) in pointed L topology, i.e., there exist smooth embedings Σ p Φ :Σ B k (p ), s.t. Φ∗ g g in L (dµ ). kr r → r k kr k → loc g 2) The complex structure of BΣk (p ) converges to the complex structure of Σ . Okr r k O r 3) F Φ converges to F weakly in W 2,2(Σ , Rn) and strongly in W 1,p(Σ , Rn). k ◦ kr r loc r loc r But we do not know whether FR is proper. We even do not know whether the ‘boundary’ of FR(ΣR)(which we will define latter) will touch ∂BR(0). So we will next extend the local limit XR := (FR, ΣR,g,p) to some maximal X = (F, Σ,g,p) such that the ‘boundary’ of F (Σ) touches ∂BR(0) and then argue the extended immersing map F is proper in BR(0). For simplification, we call the topology defined by the convergence in the last three items 1) 2) 3) by τ-topology. For example, we will say the Σk quadruple Xkr = (Fk,Br (pk),gk,pk) converges to Xr = (Fr, Σr,g,p) in τ-topology and so on. For r (0, R), we define a partial order on the set ∀ ∈ = XA = (F A, ΣA,g,p) A N and connected domains U A Σr (p ), α A s.t. Cr { |∃ ⊂ + α ⊂ α α ∀ ∈ U A BΣα (p ) and X = (F ,U A,g ,p ) converges to XA in τ-topology α ⊃ r α α α α α α } by saying XA XB if A B is infinite and U A U B for infinite γ A B. By diagonal argument, ≤ ∩ γ ⊂ γ ∈ ∩ it is not hard to show that ( r, ) is a partial order set such that every chain in ( r, ) has an upper C ≤ r r r C A≤r bound. Thus by Zorn’s lemma, there exists a maximal X = (F , Σ ,g,p) = some X in ( r, ). We r r C ≤ are going to show F :Σ Br(0) is proper for each r (0, R). r→ r ∈ r r r First of all, we define Σ¯ to be the completion of (Σ , dg) as a metric space and call ∂Σ := Σ¯ Σ the r r \ boundary of Σ . Since the area of a surface is continuous in the τ-topology, we know volg(Σ ) V < + . Hence Σr is not complete and ∂Σr = . ≤ ∞ r 6 ∅ Otherwise, Σ will be a closed surface or contains a ray. In the first case, Uα is a closed surface for R α Ar large enough. This means Σα (pα) contains a closed surface Uα and must equals to the closed surface∈ itself since it is connected. But then ΣR(p ) = U B (0) B (0), which contradicts to α α α ⊂ r ⊂⊂ R the fact Fα : Σα BR(0) is a proper immersion of an open surface. In the later case, we can choose → Σr Σr infinite many points xi on the ray such that Bi (R r)(xi) Bi (R r)(xj ) = for i = j. But by the { } 0 − ∩ 0 − ∅ ∀ 6 definition of the isothermal radius, there exist open neighborhoods U(x ) BΣ (x ) and isothermal i i0(R r) i 2u ⊃2 2− coordinates ϕi : D1(0) U(xi) such that ϕi(0) = xi and ϕi∗g = e (dx + dy ). Moreover, we know → 1 C(n,ε,V,i (R r)) i0(R r) u (z) C(n,ε,V,i (R r)) for z . Thus for z r := e− 0 − − , | | ≤ 0 − | |≤ 2 | |≤ 0 2 1 i (R r) d (z, 0) eu(tz)dt 0 − . g ≤ ≤ 2 Z0 Σr This means Bi (R r)(xi) ϕi(Dr0 (0)) and 0 − ⊃ 2 r Σ 2u πi0(R r) 4C(n,ε,V,i0(R r)) volg(Bi0(R r)(xi)) e dxdy − e− − . − ≥ D (0) ≥ 4 Z r1 So we get

r ∞ Σr V volg(Σ ) volg(Bi (R r)(xi))=+ , ≥ ≥ 0 − ∞ i=1 X again a contradiction! ¯ r r r r Then we claim Σδ := x Σ dg(x, ∂Σ ) δ is compact in Σ . Here we also need to use the { ∈ | ≥ } ¯g area bound essentially. In fact, if we choose a maximal disjoint family of closed balls B δ (xl) l I s.t. { 3 } ∈ ¯ r ¯ r ¯g ¯ r xl Σδ, then Σδ l I B 2δ (xl). So if the number of the balls is finite, then Σδ is compact. Since ∈ ⊂ ∪ ∈ 3 Ar Ar r r r Ar r Ar Xα = (Fα,Uα ),gα,pα) converges to X = (F , Σ ,g,p) in τ-topology, there exists Φα : Σ Uα , Ar p Ar → s.t. Φα ∗gα g in Lloc-topology and xαl =Φα (xl) xl in pointed Gromov-Haustorff topology. Since → Ar ¯g ¯gα → dg Cd, we know Φα (B δ (xl)) B δ (xαl) for α large enough. And by the same argument as in the ≤ 3 ⊃ 6C above paragraph, we get

¯g ¯gα volg(B δ (xl)) lim volgk (B δ (xαl)) 3 α 6C ≥ →∞ 14 J. SUN, J. ZHOU

2 δ 2 i0(R r)) 4C(n,V,i (R r),ε) lim π min , − e− 0 − α ≥ →∞ { 12C 2 } = C′(n,V,i (R r),ε,δ ), 0 − ¯g where we use lemma 3.4 in the second inequality. Since B δ (xl) l I are disjoint, we know I { 3 } ∈ | | ≤ V r ′ < + and Σ¯ is compact. C (n,V,i0(R r),ε,δ) δ − r ∞ r r r + If we denote Σδ = x Σ dg(x, ∂Σ ) > δ , then the last paragraph guarantees Σ 1 j=1∞ is a { ∈ | } { j } r r r compact exhaustion of Σ . Hence for F : Σ Br(0) to be proper in Br(0), it is enough to show r r c → F (∂Σ 1 ) Br 3 (0) for infinite j. j ⊂ j − r r r Now we show F (∂Σ 1 ) Br 3 (0) = will cause a contradiction. In fact, if x0 ∂Σ 1 s.t. y0 := j ∩ − j 6 ∅ ∃ ∈ j r 3 Ar Ar 3 3 3 F (x0) Br (0), then y0 < r j and B (y0) B + y0 (0) Br(0). Put qα = Φα (x0) Uα ∈ − j | | − j ⊂ j | | ⊂ ∈ Ar r and yα = Fα(qα). Then yα = Fα Φα (x0) F (x0) = y0 as α . So for α large enough, ◦ → → ∞1 yα B 1 (y0) and B 2 (yα) B 3 (y0) Br(0). Let Vα be the component of Fα− (B 2 (yα)) Σα containing ∈ j j ⊂ j ⊂ j ∩ q . Then i (x) i (R r), x V . Thus by the same argument as in the global case, after passing α gα ≥ 0 − ∀ ∈ α ˜ Ar Ar ˜ r ˜r ˜ r to a subsequence again, Xα = (Fα,Uα Vα,gα,pα) converges to some X = (F , Σ ,g,p) r and r ˜ r gα Ar ∪ ˜ r g ˜ r ∈ C r X X . Since B 2 (qα) Vα Uα Vα Σ and qα x0, we know B 3 (x0) Σ . But x0 ∂Σ 1 ≤ j ⊂ ⊂ ∪ → → 2j ∈ ∈ j implies d (x , ∂Σr) 1 < 3 d (x , ∂Σ˜ r), thus Σr ( Σ˜ r and Xr < X˜ r. This contradicts to Σr is g 0 ≤ j 2j ≤ g 0 maximal. r r At last, we know F : Σ Br(0) is proper for each r (0, R). Take a direct limit by diagonal argument and we get a quadruple→ XR = (F R, ΣR,g,p) such∈ that F R :ΣR B (0) is proper.  ∈CR → R 5. Application I-Leon Simon’s decomposition theorem 5.1. Estimation of the isothermal radius. We now give the estimation of the isothermal radius. The p 0 key is that the Lloc-convergence of the metric could be improved to Cloc-convergence in the blowup case (Corollary 3.3) and the isothermal radius is continuous in C0 topology. Proof of Property 1.3. We argue by contradiction. If not, there exists a V > 0, a sequence of immersing n 1 Riemannian surfaces Fk : (Σk,gk, 0) R and xk Σk B (0) satisfying µgk (Σk B1(0)) V and → ∈ ∩ 2 ∩ ≤ 2 ˜ ˜ Σ B (0) Ak dµgk = εk 0, but αk = igk (xk) = infx Σk B 1 igk (x) 0. Let Fk : (Σk,hk, 0) = k ∩ 1 | | → ∈ ∩ 2 → 1 2 n (α− (Σ x ), α− g , 0) R . Then for any fixed R> 0, we have R k k − k k k → (1) for any fixed δ > 0, 2 2 µ (Σ˜ B (0)) = (α R)− µ (Σ B (x ))R hk k ∩ R k gk k ∩ αkR k 1 2 ((1 + δ)µ (Σ B (0)) + (1 + δ− )Will(Σ ))R ≤ gk k ∩ 1 k 1 2 2 ((1 + δ)V +2(1+ δ− ) Ak dµgk )R ≤ Σ B (0) | | Z k∩ 1 1 2 ((1 + δ)V +2(1+ δ− )ε )R ≤ k (1+2δ)V R2 ≤ for k large enough, where we use the monotonicity formulae Lemma 2.2 in the first inequality; 2 (2) A˜ dµ = A 2dµ ε 4πε for k large enough; Σ˜ BR(0) k hk Σ Bα R(x ) k gk k k∩ | | k ∩ k k | | ≤ ≤ 1 1 1 ˜ 1 (3) iRhk (˜x)= αk− igk (x) 1R forx ˜ = αk− (x xk) Σk B (0) but ihk (0) = αk− igk (xk) = 1. ≥ − ∈ ∩ 2αk n That is, (F˜k : (Σ˜ k,hk, 0) R ) (n, ε, 1, (1+2δ)V, R) for k large enough. By Theorem 1.2, we know there exists a pointed Riemannian→ ∈C surface (Σ ,h , 0) and a proper immersion F : (Σ ,h , 0) Rn ∞ ∞ p ∞ ∞ → with continuous metric g such that (Σ˜ k,hk, 0) (Σ ,h , 0) in pointed L -topology and F˜k F in weak W 2,2 topology. Again, the lower semi-continuity→ ∞ property∞ of total curvature implies → 2 ˜ 2 A dµh∞ lim inf Ak dµhk =0. ∞ k Σ∞ BR(0) | | ≤ Σ˜ BR(0) | | Z ∩ →∞ Z k ∩ A GROMOV-HAUSDORFF CONVERGENCE THEOREM OF SURFACES IN Rn WITH SMALL TOTAL CURVATURE15

2 Thus Σ is a flat plane R with isothermal radius ih∞ (Σ ) = and we may assume F (x, y) = n∞2 ∞ ∞ n (x, y, 0 − ) for some constant a> 0. In fact, F (Σ ) is included in a plane P R since A = 0. Thus F : (Σ , 0) (P, 0) is a proper immersion of a∞ surface to a plane with the⊂ same dimension∞ . Since a proper∞ local diffeomorphsim→ is a covering map and P is simply connected, we know F : (Σ , 0) (P, 0) is 2 2 n ∞ → n 2 a diffeomorphsim. So we can assume Σ = R and F : R P R has the form F (x, y) = (x, y, 0 − ). Next, we observe the continuity of∞ isothermal radius in→ the⊂ special case to get a contradiction. By ˜ 0 Theorem 4.3(d), (Σk,hk, 0) (Σ ,h , 0) in Cloc topology. So,there exist local diffeomorphisms fk : → ∞ ∞ 0 ˜ C 2 Σk ˜ c 2 2 D1+2σ R Ωk B1+σ(0) Σk such that fk∗hk h = (dx + dy ) on D1+σ. Regard fk as the ⊂ → ⊃ ⊂ 2 −−→ ∞ 2 ⇒ ⇒ coordinate of Ωk and write fk∗gk = Ekdx +2Fkdxdy + Gkdy . Then we have Ek, Gk 1 and Fk 0 uniformly on D1+σ. Hence the Beltrami coefficient 2 2 (Ek Gk) +4Fk µh = − ⇒ 0 k E + G +2 E G F 2 k p k k k − k on D1+σ. So, by the existence of isothermal coordinatep [7](see also [19]), we know ihk (0) 1+ σ. A contradiction! ≥  Σ Remark 5.1. If we assume U(p) 101Br (p)(which is satisfied in the above case) in the definition of isothermal radius, then by similar argument⊂ as in the last paragraph, one can prove the isothermal radius is continuous in pointed C0 topology. This is the advantage of the isothermal radius to the injective radius. 5.2. We now give a blowup approach to Leon Simon’s decomposition theorem [27]. The idea is first arguing by blowup to show the immersions are indeed embeddings, then using the Poincar´einequality to estimate the area out of the multi-graph, and finally using the area estimate to prove the multi-graph is in fact single. During this subsection, we use Ψ(ε) to denote a function satisfying limε 0 Ψ(ε) = 0 which may Σ Σ → Σ change from line to line. We also use the notation U Br (p) to mean B(1 Ψ(ε))r(p) U B(1+Ψ(ε))r(p). ∼ − ⊂ ⊂ Theorem 5.2. For fixed R > 0 and any V > 0, there exists an ε2 = ε2(n, V ) > 0 such that for ε (0,ε2] the following holds: ∀ If∈ F : (Σ,g,p) (Rn, 0) is a Riemannian immersion proper in B (0) with → R 2 2 2 2 c (Σk BR(0)) V R , A dµg ε and F (∂Σ) BR(0), H ∩ ≤ Σ BR(0) | | ≤ ⊂ Z ∩ Σ R n then there exists an topologically disk U0(p) B R (p) with p Σ 2 (0) U0(p) such that F : U0(p) R 2 c ∼ ∈ ⊂ → is an embedding with F (∂U0) B R (0). ⊂ 2 Moreover, there exist finite many disjoint topologically disks Di contained in U0, such that Σ 2(D ) Ψ(ε), Σ 1(∂D ) Ψ(ε), iH i ≤ iH i ≤ and U0(p) i Di = graphu = (x, u(x)) x Ω \ ∪ { n| ∈ } for some function u : Ω L⊥ with u Ψ(ε), where L R is an affine 2-plane, and Ω L is a bounded domain with the→ topology type|∇ of| a ≤ disk with finite man⊂ y disjoint small disks removed. ⊂ Proof. Without loss of generality, we assume R = 1 and argue by contradiction. So assume there n are a sequence of Riemannian immersions Fk : (Σk,gk,pk) (R , 0) proper in B1(0) and satisfying 2 2 2 → Ak dµg = ε 0, (Σk B1(0)) V but the conclusion of the theorem does not hold. Σk B1(0) | | k k → H ∩ ≤ We∩ will prove the conclusion does hold for k large enough and get a contradiction. R 3 First of all, Property 1.3 implies there exists α0(V ) > 0 such that igk (x) α0(V ), x Σk B (0). ≥ ∀ ∈ ∩ 4 Then by Theorem 1.2, there exists an immersing Riemannian surface F : (Σ ,g ,p) Rn which is ∞ ∞0 ∞ → properly immersed in B 2 (0) such that (Σk,gk,pk) (Σ ,g ,p) in pointed C topology, Fk F in 3 → ∞ ∞ → ∞ pointed W 1,p and weak W 2,2 topology and

2 2 A dµg∞ lim inf Ak dµgk =0. | | ≤ k | | Σ∞ B 2 (0) →∞ Σk B 2 (0) Z ∩ 3 Z ∩ 3 16 J. SUN, J. ZHOU

n Thus F (Σ ) B 2 (0) R is a flat disk in some 2-dimensional linear subspace P . Since F is a ∞ ∞ ∩ 3 ⊂ ∞ proper immersion in B 2 (0) P , by the simply connected property of B 2 (0) P , we may assume F is 3 3 ∞ ∩ n 2 ∩n an embedding with the form F (x1, x2) = (x1, x2, 0 − ) and Fk : D 2 (0) R satisfies ∞ 3 → 0 C (D 2 ) 3 2 2 Rn 1,p β Fk∗g dx1 + dx2 and Fk F W (D 2 (0)) + Fk F C (D¯ 3 )(0) 0, −−−−−→ k − ∞k 3 k − ∞k 5 → 2 3 3 5 for β (0, 1). Moreover, we have 2 Fk(Dr) Σ (pk) and may assume 3 Fk(Dr) Σ (pk) by r< 3 k r< 5 k ∀ ∈ ⊃ 1 ⊃ 3 2 ¯ 3 taking a subsequence. So for k large enough, there exists rk < s.t. Σ (pk) Fk(Drk ) Fk(D ). S 5 k S 5 n ⊂ ⊂ Step 1. Fk : D 3 R is embedding for k large. 5 → To prove this, we also argue by contradiction(similar to [5, Lemma 3.6], see also [20, Lemma 2.1.3]). If it is not this case, then there exist two sequences of points xk = yk D 3 such that Fk(xk)= Fk(yk). { 6 }⊂ 5 β By Fk F C (D¯ 3 )(0) =: ck 0, we have k − ∞k 5 → β xk yk = (Fk(xk) F (xk)) (Fk(yk) F (yk)) ck xk yk , | − | | − ∞ − − ∞ |≤ | − | 1 β and hence xk yk − ck 0. Intuitively, this means the sequence Fk will blowup to a map degenerating| on− the| limit≤ direction→ of x y , which contradicts to the immersion assumption. More k − k precisely, we may assume xk,yk x0 D¯ 3 . Let yk = xk + rkek, where rk = xk yk and ek = 1. → ∈ 5 | − | | | Assume e e = (0, 1) S1 and define k → ∈ Fk(xk + rkx) Fk(xk) fk(x)= − . rk 2 Since Ak 0, so by the same argument as in the proof of Proposition1.3, we could assume F ∗gRn = Σk k 2 | | → 2 ⇒ ⇒ 2 ∞ Ekdx1 +2Tkdx1dx2 + Gkdx2 with Ek, Gk 1, and Tk 0 uniformly in D . Thus dFk L (D 2 ) 3. R 3 k k 3 ≤ So

Fk(xk + rke) Fk(xk) Fk(xk + rke) Fk(xk + rkek) fk(e) = | − | = | − | | | rk rk ∞ dFk L (D 2 ) ek e 0. ≤k k 3 | − | →

Furthermore, we have fk(0) = 0,

df ∞ = dF ∞ dF ∞ 3 k L (D1) k L (Drk (xk)) k L (D 1 (x0)) k k k k ≤k k 15 ≤ and 2 2 2 2 d fk = d Fk (y)dy 0, D | | D (x ) | | → Z 1 Z rk k which imply f f(x , x )= ax +bx +c in W 2,2(D ) W 1,p(D ) Cβ(D ). So, f(0) = 0 and f(e)=0 k → 1 2 1 2 1 ∩ 1 ∩ c 1 imply b = c = 0 and f(x1, x2)= ax1. But on the other hand, for a.e.x,

1 2 2 df df(x) = lim dfk dfk(x) = lim dFk dFk(xk + rkx) (dx1 + dx2), ⊗ k ⊗ k ⊗ ≥ 2 →∞ →∞ i.e., b = 0. A contradiction! Step6 2. Area estimate out of the graph. n W.O.L.G., we can assume Fk : D 3 (0) R is a conformal embedding for k large, i.e., F ∗gRn = 5 k 2u 1 1 2 2 → 2 2 e k (dx dx + dx dx ). In fact, since F ∗gRn = E dx +2T dx dx + G dx with E , G ⇒ 1, and ⊗ ⊗ k k 1 k 1 2 k 2 k k Tk ⇒ 0 uniformly in D 2 , by the existence of isothermal coordinates, there exists a ϕk : D 2 (0) D 2 (0) 3 3 → 3 such that ϕk(0) = 0 and 2uk 1 1 2 2 ϕ∗F ∗gRn = e (dx dx + dx dx ). k k ⊗ ⊗ 2uk 1 1 2 2 n If we define F˜k = Fk ϕk andg ˜k = e (dx dx + dx dx ), then F˜k : D 2 (0) R is a conformal ◦ ⊗ ⊗ 3 → 2 2 2 2 ˜ embedding with volg˜k (D (0)) = volgk (Fk(D (0)) and Ak = Ak . So Fk could be 3 3 D 2 (0) g˜k Fk(D 2 (0)) g 3 | | 3 | | replaced by F˜k and all the above arguments hold. R R A GROMOV-HAUSDORFF CONVERGENCE THEOREM OF SURFACES IN Rn WITH SMALL TOTAL CURVATURE17

11 3 Σk Take i0 = < , then Wk = Fk(Di ) B (pk). For x Di , consider the oriented Gauss map 20 5 0 ∼ i0 ∀ ∈ 0 2uk ∂Fk ∂Fk 1 Gk(x)= e− ∂x (x) ∂x (x), and let fk(x)= Gk(x) Vk , where Vk = πi2 D Gk(x)dx. Then 1 ∧ 2 | − | 0 i0 Σk f (x) Σk G (x)= A (x). R |∇ k|gk ≤ |∇ k|gk | k|gk 1 So, for η > 0, t (η /2, η ) such that t is regular value of f (i.e., Γ = f − (t ) consists of some ∀ k ∃ k ∈ k k k k k k k circles and arcs in Di0 ), and

ηk 1 2 1 (Γk) ηk fk = t dt H ≤ ηk ηk H { } − 2 Z 2 ηk 4 Ak gk 1 |Σ | (x) fk = t dt ≤ ηk ηk k fk g H { } Z 2 |∇ | k

4 2uk = Ak gk (x)e dx η ηk k f η | | Z 2 ≤ k≤ k 4 1 1 2 2uk 2 2 2uk 2 ( Ak gk e ) ( 1 e dx) ≤ ηk D | | D Z i0 Z i0 8√π i0εk. ≤ ηk Let S = x D f (x) >t . Then by the Poincar´einequality [10, section 7.8, page164], we get k { ∈ i0 | k k} 2 2 1 1 4√2πi 1 4√2πi 2 2 0 2 2 0 ( fk fk ) 1 ( fk ) 1 εk, D | − Sk S | ≤ S 2 D |∇ | ≤ S 2 Z i0 | | Z k | k| Z i0 | k| where we use the conformal invariance of the Dirichlet integral in the last inequality. But we also have,

1 2 1 1 1 ( f f ) 2 f 2 f 2 t S 2 f 2 , | k − S k| ≥k S kkL (Sk) −k kkL ≥ k| k| −k kkL ZSk | k| ZSk | k| ZSk and

1 2 1 2 2 fk L (Sk) ( Gk(x) 2 Gk ) k k ≤ D | − πi D | Z i0 0 Z i0 2 1 4 n(n 1)i ( G (x) ) 2 4 n(n 1)i ε , ≤ − 0 |∇ k | ≤ − 0 k Di0 p Z p where we use the conformal invariance of the Dirichlet integral again. Thus we have

2 1 4√2πi 4 n(n 1)i0 2 0 tk Sk 1 εk + − εk, | | ≤ S 2 √πi0 | k| p √εk thus if we take ηk = √εk, then tk ( , √εk) and ∈ 2 1 S (64+8 n(n 1)π)ε 2 i2. | k|≤ − k 0 1 2 p So, when taking ηk = εk , we have the length estimate and the area estimate:

1 1 1(Γ ) 8√πε 2 i , S (64+8 n(n 1)π)ε 2 i2 H k ≤ k 0 | k|≤ − k 0 1 p Furthermore, since G (x) V = f (x) t ε 2 √2 for x D S , W F (S ) could be | k − k| k ≤ k ≤ k ≪ ∀ ∈ i0 \ k k\ k k written as a multi-graph with gradient small. More precisely, if we define Lk = [Vk] to be the linear space 2 n n corresponding to Vk Λ (R ) and πk : R Lk the orthogonal projection, Ωk′ = πk(Wk Fk(Sk)), then π is local diffeomorphism∈ when restricted on→ W F (S ): \ k k\ k k If not, there must exists x Wk Fk(Sk) and e1(x) [Gk(x)] kerdπ = [Gk(x)] Lk⊥, that is, for any orthogonal basis e0,e0 of L , ∈e ,e0\ = e ,e0 = 0. Choose∈ any e∩ [G (x)] such that∩ [G (x)] = [e e ], 1 2 k h 1 1i h 1 2i 2 ∈ k k 1∧ 2 then e e ,e0 e0 = 0, and G(x) V = e e 2 + e0 e0 2 = √2. h 1 ∧ 2 1 ∧ 2i | − k| | 1 ∧ 2| | 1 ∧ 2| p 18 J. SUN, J. ZHOU

Hence there is a multi-function v′ : Ω′ L⊥ with gradient v′ Ψ(t ) = Ψ(ε ) such that W = k k → k |∇ k|≤ k k k graphvk′ . But when noticing that 1 1 ∂ ∂ Vk = Gk G = = V 2 2 ∞ ∞ πi0 Di → πi0 Di ∂x1 ∧ ∂x2 Z 0 Z 0 (by the control convergence theorem), we know that Graphvk′ could be written as a Graph over L = [V ] ∞ ∞ with gradient v′ + Vk V = Ψ(εk). That is: ≤ |∇ k| | − ∞| 2 Wk Fk(Sk) = Graph of vk :Ωk V V ⊥, vk Ψ(εk), \ ⊂ ∞ → ∞ |∇ |≤ where Ω2 is a bounded planner domain contained in Ω1 = π(W ) D k k k ∼ i0 The domain Wk Fk(Sk) may not be connected since there are some arcs in Γk with ends on ∂Wk. But \ 1 1 1 2 2 2 when noticing (Γk) 8√πε i0, Sk (64+8 n(n 1)π)ε i0 and Fk F , we know Wk Fk(Sk) H ≤ k | |≤ − k → ∞ \ contains a unique large connected component C′ W F (S ) such that k p∼ k\ k k 2 2 2 (1 Ψ(ε ))πi (C′ ) (1 + Ψ(ε ))πi . − k 0 ≤ H k ≤ k 0 W.L.O.G., we can assume Γk does not contain any arc and Ck′ = Wk Fk(Sk). Step 3. The multi-graph is in fact single valued. \ 2 n Consider Ωk := π(Ck′ ), where the orthogonal projection π : R L is local diffeomorphism when α n → ∞ α 2 restricted on Ck′ . Since Fk F in C (Di0 ; R ), we know π Fk Id Di in C (Di0 ; R ) and hence → ∞ ◦ → | 0 2 3 Ωk = π(Ck′ )= π Fk(Di0 Sk) D(1 Ψ(εk ))i0 π Fk(Sk)=:Ωk. ◦ \ ⊃ − \ ◦ Since π is local diffeomorphism when restricted to C′ , we know ∂π F (S ) π(∂F (S )) = π(Γ ) and k ◦ k k ⊂ k k k 1 1(∂π F (S )) 1(π(Γ )) 1(Γ ) 8√πε 2 i , H ◦ k k ≤ H k ≤ H k ≤ k 0 thus we may assume 3 Ωk D(1 Ψ(εk))i0 i Ik di =: Ωk, ⊃ − \ ∪ ∈ 1 where di i Ik is a finite set of disjoint closed topological disks in D(1 Ψ(εk))i0 such that Σi Ik (∂di) 1{ } ∈ − ∈ H ≤ 2 2 8√πεk i0 and so Σi Ik di 16εki0 by the isoperimetric inequality. ∈ 1| |≤ Now, let Ck = π− (Ωk) Ck′ . Then also because π is local diffeomorphism on Ck′ , there are finite ∩ 1 1 2 many disjoint closed topological disks Di i Jk Wk with Σi Ji (∂Di) (1 + Ψ(εk))(8√πεk i0) such that { } ∈ ⊂ ∈ H ≤

Ck = Ck′ i Jk Di = Wk i Jk Di \ ∪ ∈ \ ∪ ∈ and π(∂Ck)= π( i J ∂Di)= ∂Ωk, ∪ ∈ k that is, π : C Ω is a proper local diffeomorphism and hence a covering map. But on the other hand, k → k C C′ implies k ⊂ k 2(C ) (1 + Ψ(ε ))πi2 H k ≤ k 0 but 2 2 2 Ωk = D(1 Ψ(εk))i0 i Ik di (1 Ψ(εk))πi0 16εki0 = (1 Ψ(εk))πi0. | | | − \ ∪ ∈ |≥ − − − Thus π : Ck Ωk must be a single cover, i.e., vk is single valued when restricted on Ωk. Moreover, → 2 the uniform convergence of the metrics induced by Fk : Di0 Wk again implies Σi Jk (Di) 1 → ∈ H ≤ 3 2 2 (1 + Ψ(εk)) 8√πεk i0. Letting U (p )= W , u = v we get the conclusion!  0 k k k k|Ωk From the Decomposition theorem, we can get the following “area drop” property.

Corollary 5.3 (Density Drop). For fixed any V > 0, there exists an ε2 = ε2(n, V ) > 0 such that for ε (0,ε2] the following holds: ∀ If∈ F : (Σ,g,p) (Rn, 0) is a Riemannian immersion proper in B (0) with → R 2 2 2 2 (Σk BR(0)) V R and A dµg ε , H ∩ ≤ Σ BR(0) | | ≤ Z ∩ A GROMOV-HAUSDORFF CONVERGENCE THEOREM OF SURFACES IN Rn WITH SMALL TOTAL CURVATURE19

then 2 R (Σ 2 (0)) H 1+Ψ(ε). R 2 π( 2 ) ≤ Proof. Since the Gradient of the graph map and the measure outside the graph are both well estimated, we get the result.  Remark 5.4. With this dropped area density, by a similar argument as in the first step of the proof of the Allard Regularity theorem–the Lipschitz Approximation–and Reifenberg’s Topological disk theorem, 1 1 Ψ (ε)R Ψ (ε)R we know there exists some Ψ (ε) = min Ψ(ε) 2 ,ε , such that for any ξ Σ 1 (p), Σ 1 (ξ) is 1 219 { } ∈ a topological disk. This means, in the small scale of Ψ1(ε)R, there is no holes caused by intersecting Σ with an extrinsic ball BΨ1(ε)R(0). 6. Application II-Helein’s´ convergence theorem 6.1. H´elein’s convergence theorem was first proved by H´elein [12]. An optimal version of the theorem was stated in [14] as following: 2,2 n Theorem 6.1. Let fk W (D, R ) be a sequence of conformal immersions with induced metric gk = 2uk ∈ e δij and satisfy

2 8π for n =3, Af dµg γ<γn = | k | k ≤ 4π for n 4. ZD ( ≥ 2,2 n Assume also that µgk (D) C and fk(0) = 0. Then fk is bounded in W (Dr, R ) for any r (0, 1), and there is a subsequence≤ such that one of the following two alternatives holds: ∈ 2,2 n (a) uk is bounded in L∞(Dr) for any r (0, 1), and fk converges weakly in Wloc (D, R ) to a conformal 2,2 n ∈ immersion f Wloc (D, R ); (b) u and∈ f 0 locally uniformly on D. k → −∞ k → 2,2 2 n Corollary 6.2. Let fk W (S , R ) be a sequence of immersions with gk = dfk dfk and satisfy Will(f ) C. If f (x )=0∈ for some x S2 and ⊗ k ≤ k 0 0 ∈ vol (S2) V. gk ≡ Then there exist a subsequence(still denoted as f ) and a sequence of M¨obius transformations φ (S2) k k ∈M such that f˜ = f φ converges weakly to some immersion f W 2,2(S2 S, Rn) in W 2,2(S2 S, Rn), where k k ◦ k ∈ loc \ loc \ S = x S2 lim lim inf A 2dµ γ f˜k g˜k n { ∈ | r 0 k g0 | | ≥ } → →∞ ZBr (x) is the finite set of singular points in S2. Proof. Let µ = 2xf (S2) and ν = µ x A 2. Then ν µ , µ (Rn) = V and ν (Rn) 4π + C by k H k k k | k| k ≪ k k k ≤ the Gauss-Bonnet formulae. So after passing to subsequences, µk ⇀ µ and νk ⇀ ν as Radon measures 2 2 (ν µ). Let dk = diam(fk(S )) = sup p q p, q fk(S ) . We claim: ≪ 1 {| − || ∈ } πC − 2 (1) dk d := min 1, 2(1 + V ) (2) spt≥µ contains{ at least infinite points,} 2 d  (3) for pk fk(S ) and r< 2 , ∀ ∈ µ (B (p )) 6V k r k θ := +2C. πr2 ≤ πd2 d 1 (4) p sptµ and r (0, 2 ), s.t. ν(Br(p)) 2 ε0(θ), where ε0(θ) is the small constant defined in Proposition∃ ∈ 1.3. ∈ ≤ In fact, since 0 f (S2)= sptµ , if d 1, then by the monotonicity formulae, δ > 0, ∈ k k k ≤ ∀ 2 2 2 2 V (fk(S )) Bdk (0) (fk(S )) B1(0) 1 2 = H 2∩ (1 + δ)H ∩ +(1+ )Will(fk) πdk πdk ≤ π 4δ (1 + δ)V 1 +(1+ )C. ≤ π 4δ 20 J. SUN, J. ZHOU

1 Taking δ = 2 , we get dk d(C, V ). Moreover, if sptµ = p1,p2,...pl is a finite set, then for each p sptµ, by the monotonicity≥ formulae again, we know for r{ 1, } i ∈ ≤ 2 n l l V = lim µk(fk(S )) lim µk(R i=1 Br(pi)) + lim Σi=1µk(Br(pi)) k ≤ k \ ∪ k →∞ →∞ →∞ n l l µk(Br(pi)) 2 = µ(R i=1 Br(pi)) + lim Σi=1 πr \ ∪ k πr2 →∞ 2 l µk(B1(pi)) 1 πr lim Σi=1 (1 + δ) +(1+ )Will(fk) ≤ k { π 4δ } →∞ V 1 (1 + δ) +(1+ )C πlr2, ≤ π 4δ letting r 0 we get V = 0. A contradiction! (3) is also a simple corollary of the monotonicity formulae 1→ for δ = 2 . 4π+C N d Now, let N := [ ] + 1 and take pi sptµ. Then r > 0(we can assume r < ) s.t. ǫ0/2 i=1 2 { } ⊂N ∃ n Br(pi) Br(pj ) = , 1 i = j N. Hence Σi=1ν(Br(pi)) ν(R ) 4π + C and there exists ∩ ∅ ∀ ≤ 46π+C ≤ ε0 ≤ ≤ 1 i0 N, s.t. ν(Br(pi0 )) N 2 . ≤ ≤ ≤ ≤ 2 ε0 Moreover, since pi sptµ, there exists pk fk(S ) s.t. pk pi 0, thus νk(B r (pk)) ν(Br(p)) 0 ∈ ∈ | − 0 | → 2 ≤ ≤ 2 for k large enough. Similar to (1), if we define d (p ) = sup p p p f (S2) , then d (p ) d hence k k {| − k|| ∈ k } k k ≥ 2 r 1 r fk : S B (p) B d (p) properly for k large enough. So, by Proposition 1.3, for x fk− (B (pk)), → 2 ⊂ 2 ∀ ∈ 4 the isothermal radius r i (x) α (θ) =: r > 0. gk ≥ 0 2 1 i 2 1 0 i 0 r1 Especially, we could choose x f − (B r (p )) such that f (x ) = p and f (x ) f (x ) = k i=0 k 4 k k k k k k k k 2 for i =1, 2. { } ⊂ | − | i 2 2 2 Fix three different points x0 i=0 S , there exists a unique M¨obius transformation φk (S ) s.t. i i { } ⊂ 2 n ∈M φ (x )= x ,i =0, 1, 2. Consider f˜ = f φ : S R andg ˜ = φ∗g . Let k 0 k k k ◦ k → k k k S = x S2 lim lim inf A 2dvol γ . f˜k g˜k n { ∈ | r 0 k g0 | | ≥ } → →∞ ZBr (x) Then S is a finite set and by Theorem 6.1, f˜ converges to some f weakly in W 2,2(S2 S, Rn) such that k loc \ either f is an immersion or f(S2 S) p. But by the choice of xi , we know xi / S,i = 0, 1, 2 and \ ≡ 0 0 ∈ f˜ (x0) f˜ (xi ) = r1 > 0,i =1, 2, thus f(S2 S) = p and f : S2 S Rn is an immersion.  | k 0 − k 0 | 2 \ 6 \ → References

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Jianxin Sun: Academy of Mathematics and Systems Science, CAS, Beijing 100190, P.R. China. Email:[email protected]

Jie Zhou: Academy of Mathematics and Systems Science, CAS, Beijing 100190, P.R. China. Email:[email protected]