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It is straightforward to extend Theorem 3 to hold for certain non-compact N, see [50, §3] for one such formulation. Theorems 2 and 3 are the four-dimensional analogue of the well-known curvature estimate of Schoen [39] for minimal surfaces in three-dimensions. Note that by the work of Schoen–Simon–Yau [37], such an estimate was previously known to hold where C depended on an upper bound for volume of M 3 in small balls. A slight modification of the proof of Theorem 1 yields a structure theorem for finite index minimal hypersurfaces in R4, analogous to the well-known results of Gulliver, Fischer–Colbrie, and Osserman [20, 23, 34]. Recall that a complete, two-sided, im- mersed minimal hypersurface M n → Rn+1 has finite Morse index if ∞ index(M) := sup {dim V : V ⊂ Cc (M),Q(f,f) < 0 for all f ∈ V \ {0}} < ∞, 2 2 2 where Q(f,f)= M |∇f| − M |AM | f . Theorem 4. AR complete, two-sided,R minimal immersion M 3 → R4 has finite Morse 3 index if and only if it has finite total curvature M |AM | < ∞. We remark that Tysk [49] proved the sameR statement for a complete, two-sided, minimal immersion M n → Rn+1 (for 3 ≤ n ≤ 6) under the assumption that M has Euclidean volume growth. Theorem 4 has strong consequences on the structure of M near infinity. We recall the following definition. Definition 5 ([43, Section 2]). Suppose n ≥ 3, M n → Rn+1 is a complete minimal immersion. An end E of M is regular at infinity if it is the graph of a function w over a hyperplane Π with the asymptotics n 2−n −n −n w(x)= b + a|x| + cjxj|x| + O |x| , j=1 X for some constants a, b, cj , where x1, · · · ,xn are the coordinates in Π. By [4, 43, 44, 49] (see also [25, Appendix A]), if M n → Rn+1 is a minimal immersion with finite total curvature, then each end of M is regular at infinity. Moreover, by [27] a two-sided minimal immersion with finite Morse index has finitely many ends. Combined with Theorem 4, this yields the following result. Corollary 6. Suppose M 3 → R4 is a complete, two-sided, minimal immersion with finite Morse index. Then M has finitely many ends, each of which is regular at infinity. In particular, M has cubic volume growth, i.e.
Vol (Br(0) ∩ M) sup 3 < ∞. r→∞ r In fact, by [25], we can bound the volume growth rate linearly in terms of the Morse index Vol (B (0) ∩ M) sup r ≤ 6 (index(M) + 1) . 4 3 r→∞ 3 πr These results should be relevant to the study of minimal hypersurfaces with bounded index in a four-manifold (cf. [12, 46]). STABLE MINIMAL HYPERSURFACES IN R4 3
1.1. Idea of the proof of Theorem 1. Consider a non-flat, complete, two-sided stable minimal hypersurface M 3 → R4. It turns out to be no loss of generality to assume that M is simply connected and of bounded curvature. It is well-known (see, e.g. [26]) that M admits a positive Green’s function of the Laplacian operator denoted here by u. (In other words, M is non-parabolic.) The central quantity considered in the proof of Theorem 1 is
F (t) := |∇u|2 ZΣt
−1 where Σt = u (t). (Quantities of this type have been considered in several contexts including [33, 13, 15, 16, 31, 2, 1].) A simple computation shows that F (t) = 4πt2 when M = R3. For general M, we can view any estimate of F (t) as t → 0 as certain kind of control on the growth rate of M near infinity. It is straightforward to show that F (t) satisfies the a priori estimate F (t)= O(t) as t → 0 (see Lemma 13). An important observation is that if we can upgrade this to the sharp estimate F (t)= O(t2), then we can conclude that M is flat. At a heuristic level, if we were to pretend that u ∼ r−1 and |∇u|∼ r−2 (this holds 3 4 −4 2 on R ), we would have F (t) ∼ t |∂Bt−1 | = r |∂Br|. Thus F (t) = O(t ) can be 2 thought of as the estimate |∂Br| = O(r ) which would suffice to show M is flat by work of Schoen–Simon–Yau [37]. In fact, a closely related argument actually works: we can consider f = ϕ ◦ u in the L3-version of the Schoen–Simon–Yau estimates (see Proposition 25) and use the co-area formula combined with a log-cutoff near the pole and the end to see that F (t)= O(t2) implies that M is flat. It thus remains to explain how to show F (t) = O(t2). Our strategy is to find two (competing) estimates relating F (t) and the quantity
2 A(t)= |AM | . ZΣt
One can view A(t) as measuring the bending of M near infinity in a certain sense. Our first estimate follows by extending a recent argument of Munteanu–Wang who obtained [31] a sharp monotonicity formula for the quantity F (t) on a 3-manifold with non-negative scalar curvature by a clever application of Stern’s rearrangement of the Bochner formula [47] (allowing them to leverage Gauss–Bonnet on the level sets Σt). 2 In the present situation, M actually has non-positive scalar curvature RM = −|AM | , so this comes in with a bad sign when applying their method. Moreover, our a priori bound F (t) = O(t) causes complications with a crucial step in their argument. How- ever, keeping track of this bad sign and developing a delicate regularization procedure to control this limt appropriately (cf. Remark 17), we find
1 t 1 1 F (t) ≤ O(t2)+ t A(s) ds + t3 s−2A(s) ds 4 4 Z0 Zt 4 OTIS CHODOSH AND CHAO LI
(see Proposition 16). To obtain an estimate for F (t) we thus need to estimate A(t) appropriately. We achieve this by considering1 the stability inequality (1) with f = 1 |∇u| 2 ϕ(u). Using Stern’s rearrangement of the Bochner formula and the co-area for- mula, we obtain ∞ 8π ∞ 4 ∞ ϕ(s)2A(s) ds ≤ ϕ(s)2 ds + ϕ′(s)2F (s) ds, 3 3 Z0 Z0 Z0 which yields
t 1 4 1 t A(s) ds + t3 s−2A(s) ds ≤ O(t2)+ t3 s−4F (s) ds 3 Z0 Zt Zt after a judicious choice of the test function (see Proposition 11 and Corollary 14). Putting these inequalities together we obtain
1 1 F (t) ≤ O(t2)+ t3 s−4F (s) ds. 3 Zt 1 Because the constant 3 is < 1, it is possible to “absorb” the integral into the left-hand side, yielding F (t)= O(t2). As explained above, this implies that M is flat. We emphasize that both of the competing inequalities described above rely on Stern’s Bochner formula, and thus use the three-dimensionality of M in a strong way. In particular, Stern’s version of the Bochner formula as used in the stability inequality preserves more of the factor in front of A(s) than the “standard” Bochner formula would (basically, this is due to the use of Gauss–Bonnet). Extra loss at this step would 1 make the constant 3 larger and could cause issues with the “absorption” step.
1.2. Organization of the paper. In Section 2 we collect basic properties of the Green’s function on a stable minimal hypersurface. In Section 3 we combine Stern’s Bochner formula with the stability inequality and in Section 4 we extend the work of Munteanu–Wang and combine this with the previous section to prove that F (t) = O(t2). In Section 5 we use this to prove Theorems 1, 2, and 3. Appendix A recalls the one-ended (finite-ended) property of stable (finite index) minimal hypersurfaces, Appendix B recalls Yau’s Harnack and Michael–Simon Sobolev inequality in this set- ting, Appendix C recalls properties of the nodal/critical set of a harmonic function, and Appendix D proves a L3 version of the Schoen–Simon–Yau estimates.
1.3. Acknowledgements. We are grateful to Rick Schoen and Christos Mantoulidis for their interest and encouragement as well as to Alice Chang for a helpful discussion. O.C. was supported by an NSF grant (DMS-2016403), a Terman Fellowship, and a Sloan Fellowship. C.L. was supported by an NSF grant (DMS-2005287).
1This part of the argument is inspired by the work of Schoen–Yau [41] who considered f = |∇u|ψ and used the standard Bochner formula in (1) to prove that M does not admit finite energy harmonic functions. STABLE MINIMAL HYPERSURFACES IN R4 5
2. The Green’s function on a stable minimal hypersurface It is well-known (cf. [26, Theorem 10.1]) that when n> 2 a stable minimal hypersur- face in Rn+1 is non-parabolic (i.e., admits a positive Green’s function). We recall this fact below as well as establishing several useful facts about the behavior of the level sets of such a Green’s function. Proposition 7. For n > 2, suppose that M n → Rn+1 is a complete, connected, sim- ply connected, two-sided, stable minimal immersion with uniformly bounded curvature ∞ |AM | ≤ K. Then, for p ∈ M, there exists u ∈ Cloc(M \ {p}) with the following properties: (1) u is harmonic, ∆u = 0, on M \ {p}, (2) u> 0 on M \ {p} and inf u = 0, 1 −1 1 −2 (3) u(x) = 4π (1 + o(1))dM (x,p) , |∇u|(x) = 4π (1 + o(1))dM (x,p) , as well as ∇u 1 −3 ∇|∇u|, |∇u| = 2π (1 + o(1))dM (x,p) as x → p, 2 (4) ifD K is a compactE set containing p, then M\K |∇u| < ∞, (5) u(x) → 0 as d (x,p) →∞, M R (6) for all s ∈ (0, ∞), the set Ωs := {u ≥ s} ∪ {p} is compact, and −1 (7) if s ∈ (0, ∞) is a regular value of u then ∂Ωs = u (s) := Σs is a closed connected surface in M. Proof. If M is a flat Rn, then we can take u to be the standard Green’s function on Rn. As such, we assume below that M is not flat. We construct a sequence of approximations to u following [26, §2]. Choose an ex- haustion Ω1 ⊂ Ω2 ⊂···⊂ M with Ωi a pre-compact open set, p ∈ Ωi, and ∂Ωi smooth. ∞ ¯ There exists ui ∈ Cloc(Ωi \ {p}) so that (1’) ∆ui =0onΩi \ {p}, (2’) ui > 0 on Ωi and ui =0 on ∂Ωi, and 1 −1 1 −2 (3’) ui(x) = 4π (1 + o(1))dM (x,p) , |∇ui|(x) = 4π (1 + o(1))dM (x,p) , as well as ∇|∇u |, ∇ui = 1 (1 + o(1))d (x,p)−3 as x → p. i |∇ui| 2π M This is simplyD the statementE of existence of a Green’s function on a compact manifold with boundary. Note that ui(x) ≤ uj(x) for i < j and x ∈ Ωi \ {p}. Indeed, by (2’), for any δ > 0, it holds that ui(x) ≤ (1+δ)uj (x) for x ∈ ∂Bε(p) for all ε> 0 sufficiently small (depending on δ). Since 0 = ui(x) < uj(x) for x ∈ ∂Ωi, we thus find ui ≤ (1 + δ)uj on Ωi \ Bε(p). Sending δ, ε → 0, the inequality ui(x) ≤ uj(x) follows. Set
µi := max ui(x). x∈∂Ω1 The maximum principle implies that
u1 ≤ ui ≤ u1 + µi (2) on Ω1. We have seen that {µi}i∈N is increasing.
Claim (cf. [26, Theorem 2.3]). The sequence {µi}i∈N is bounded above. 6 OTIS CHODOSH AND CHAO LI
Proof of the claim. Assume that µi →∞. The Harnack inequality (cf. Proposition 22) −1 and interior estimates implies that up to passing to a subsequence, µi ui converges in ∞ ∞ Cloc(M \ {p}) to some non-negative harmonic function u ∈ Cloc(M \ {p}). Because we have assumed that µi →∞, then we find that 0 ≤ u ≤ 1 on Ω1. On the other hand, the maximum principle (and Dirichlet boundary conditions for ui) implies that max ui = µi, Ωi\Ω1 so u ≤ 1 on M \ Ω1. The maximum principle then implies that u ≡ 1 on M \ {p}. 0,1 ∞ Choose fi ∈ C (M) ∩ C (Ω¯ i \ Ω1) so that
∆fi =0 inΩi \ Ω1 fi =1 inΩ1 fi =0 in M \ Ωi. −1 The maximum principle implies that µi ui ≤ f on Ωi \ Ω1, so we see that fi → 1 0 ∞ in Cloc(M) ∩ Cloc(M \ Ω1). Taking fi in the stability inequality (1) we find (cf. [26, Theorem 10.1]) 2 2 2 |AM | fi ≤ |∇fi| = − Dνfi ZM ZΩi\Ω1 Z∂Ω1 for ν the outwards pointing unit normal to Ω1. Letting i → ∞, since fi converges to 1, we find that 2 |AM | = 0, ZM a contradiction since we assumed that M was not flat.
Write µi → µ∞. The claim, the Harnack inequality (cf. Proposition 22), and interior estimates allow us to pass to a subsequence so that ui converges to a harmonic function ∞ u in Cloc(M \ {p}). Note that u> 0 by the maximum principle. By (2) we have u1 ≤ u ≤ u1 + µ∞. After shifting u so that inf u = 0, we see that we have established (1)-(3). (The gradient estimate in (3) follows because u − u1 is bounded near p and thus extends across p.) We now establish property (4). Consider the solution to the following problem
∆wi =0 inΩi \ Ω1 wi =0 on ∂Ωi (3) wi = u on ∂Ω1
By the maximum principle (and since ui is increasing to u), ui ≤ wi ≤ u on Ωi \ Ω1. ∞ Thus, we find that wi converges to u in Cloc(M \ Ω1). Furthermore, we have that
2 2 |∇wi| ≤ |∇w1| ZΩi ZΩ1 for i ∈ N. Passing this inequality to the limit, we have proven (4). We now consider property (5) (cf. [33, Remark 2.4(d)]). We claim that for any (intrinsically) diverging sequence {xi}i∈N ⊂ M it holds that u(xi) → 0. Choose ϕ ∈ STABLE MINIMAL HYPERSURFACES IN R4 7
∞ C (M) so that ϕ ≡ 0 in Ω2 and ϕ ≡ 1 on M \ Ω3. Consider the function ϕwi where 0,1 wi is as in (3). Note that ϕwi ∈ Cc (M). By the Michael–Simon–Sobolev inequality (cf. Proposition 23) we have n−2 2n n n−2 2 2 2 2 (ϕwi) ≤ C ϕ |∇wi| + wi |∇ϕ| . ZM ZM Because wi has uniformly bounded Dirichlet energy and ∇ϕ is compactly supported ∞ we can let i →∞ (recalling that wi converges to u in Cloc(M \ Ω1) to find n−2 2n n (ϕu) n−2 < ∞. ZM In particular, n−2 n 2n lim u n−2 = 0. (4) i→∞ ZM\Ωi ! We now assume (for contradiction) that lim sup u = a ∈ (0, ∞). i→∞ M\Ωi
If this holds, we can find a (intrinsically) diverging sequence {xi}i∈N ⊂ M with u(xi) → a as i →∞. Passing to a subsequence we can assume that
dM (xi, Ωi) →∞ as i → ∞. Because M has uniformly bounded curvature, Schauder estimates yield bounds on all derivatives of curvature. Thus, we can pass (M, gM ,xi) to the limit (M,˜ g,˜ x˜) in the pointed Cheeger–Gromov sense (where gM is the induced metric on M). Note that by the Harnack inequality (cf. Proposition 22) and interior estimates we can also pass the harmonic function u to a pointed limitu ˜ (passing to a further subsequence). Note thatu ˜ is harmonic,u ˜(˜x) = a andu ˜ ≤ a. Thus, the maximum principle implies thatu ˜ ≡ a. On the other hand, (4) implies thatu ˜ ≡ 0. This is a contradiction, completing the proof of property (5). Properties (3) and (5) imply property (6). We now consider property (7). For s a regular value, Ωs = {u ≥ s} is a compact set −1 with smooth boundary Σs = u (s) a closed (regular) surface.
Claim. For s ∈ (s0, ∞) a regular value of u, M \ Ωs has exactly one component and it is unbounded.
Proof of the claim. Assume that M \ Ωs has a bounded component Γ. Note that 0 < u
To finish the proof of (7), if Σs were disconnected, then by the claim we can con- struct a loop with non-trivial algebraic intersection with one of the components of Σs (concatenate a path connecting two of the components of Σs inside of Ωs with one outside of Ωs). This contradicts the assumed simple connectivity of M. 8 OTIS CHODOSH AND CHAO LI
2.1. Integration over level sets of the Green’s function. Given M n → Rn+1 a complete, non-compact, connected, simply connected two-sided stable minimal immer- sion with uniformly bounded curvature. For p ∈ M fixed, consider the Greens’ function ∞ u ∈ Cloc(M \ {p}) constructed in Proposition 7. Set R the regular values of u and S −1 the singular values of u. Recall that we have defined Σs := u (s). Below we will use this notation even when s ∈ S is a singular value. Lemma 8. The set of regular values R is open and dense in (0, ∞). Proof. Proposition 7 implies that u : M \ {p} → (0, ∞) is proper. This easily is seen to imply openness of R. Density follows from Sard’s theorem as usual. For any s ∈ (0, ∞) we set ∗ Σs = {x ∈ M : u(x)= s, |∇u|(x) > 0}. ∗ Note that Σs is a smooth (possibly incomplete) hypersurface in M. By Proposition ∗ n−1 ∗ 24, dimHaus(Σs \ Σs) ≤ n − 2. In particular, we see that H (Σs) < ∞ and for any ∞ function f ∈ Cloc(M), n−1 fdH ⌊Σs = f ∗ Z ZΣs where the right hand side is taken with respect to the induced Riemannian volume ∗ form on Σs. 0 Lemma 9. For f ∈ C (M \ {p}), the function s 7→ ∗ f is continuous. loc Σs ∗ n−2+ηR Proof. Cover Σs \ Σs by balls Bri (xi) with i ri < ε and ri ≤ 1. It is clear that
δ 7→ P f Σ \∪ B (x ) Z s+δ i ri i is continuous at δ = 0. Furthermore, by Proposition 24, we have
n−1 n−1 f ≤ C H (Bri (xi)) ≤ C ri ≤ Cε, Σs+δ∩∪iBri (xi) i i Z X X where the constant C is independent of δ small. Putting these facts together, the assertion follows. We now consider the continuous functions
F (s) := |∇u|2 (5) ∗ ZΣs 2 A(s) := |AM | , (6) ∗ ZΣs defined for s ∈ (0, ∞). Lemma 10. The function F (s) is locally Lipschitz on (0, ∞). Proof. For a fixed compact subset K ⊂ (0, ∞), consider s −1 Consider the region Ωs,t = u ([s,t]). Observe that Ωs,t is a compact region with smooth boundary Σs ∪ Σt. Write η for the outwards pointing unit normal to Ωs,t and ∇u ∇u note that η = |∇u| along Σs and η = − |∇u| . We have F (s) − F (t)= h|∇u|∇u, ηi Z∂Ωs,t 1 = lim (|∇u|2 + δ) 2 ∇u, η δ→0 ∂Ωs,t Z D E 1 = lim div (|∇u|2 + δ) 2 ∇u δ→0 Ωs,t Z 1 = lim − 1 (|∇u|2 + δ)− 2 ∇|∇u|2, ∇u . δ→0 2 ZΩs,t Note that 1 (|∇u|2 + δ)− 2 ∇|∇u|2, ∇u ≤ |∇|∇u|2|≤ 2|∇u||D2u|, using the Kato inequality. From this, we find t 2 n−1 |F (s) − F (t)|≤ |∇u||D u|≤ C H (Στ )dτ, ZΩs,t Zs using the co-area formula (where C = C(M, g, u, K)). Using Proposition 24, to bound the volume of Στ , the assertion follows (the Lipchitz estimate for s,t possibly singular follows from the above estimate and continuity of F (t) proven in Lemma 9). 3. Stern’s Bochner formula and stability Recently, Stern has discovered [47] that one can combine the Bochner formula with the Schoen–Yau rearrangement [42] of the Gauss equation to relate the scalar curvature of a three-manifold to the behavior of a harmonic function on that manifold. In this section we consider this idea in the context of the stability operator for a stable minimal 3-dimensional hypersurface (cf. [41]). We consider M 3 → R4 a complete, connected, simply connected, two-sided, stable minimal immersion with uniformly bounded curvature. For p ∈ M fixed, consider the ∞ Greens’ function u ∈ Cloc(M \ {p}) constructed in Proposition 7. Recall the definition of F (s), A(s) in (5) and (6). By Lemma 9, F (s), A(s) are continuous on (0, ∞). 0,1 Proposition 11. For any function ϕ ∈ Cc ((0, ∞)) it holds ∞ 8π ∞ 4 ∞ ϕ(s)2A(s) ds ≤ ϕ(s)2 ds + ϕ′(s)2F (s) ds. 3 3 Z0 Z0 Z0 Proof. Consider the following regularization of the square root of the gradient of u 2 1 eδ = (|∇u| + δ) 4 . ∞ Note that eδ ∈ Cloc(M \ {p}) and 1 −3 2 ∇eδ = 4 eδ ∇|∇u| , as well as 1 −3 2 3 −7 2 2 ∆eδ = 4 eδ ∆|∇u| − 16 eδ |∇|∇u| | . 10 OTIS CHODOSH AND CHAO LI Applying the Bochner formula, we find 1 −3 2 2 3 −7 2 2 ∆eδ = 2 eδ (|D u| + RicM (∇u, ∇u)) − 16 eδ |∇|∇u| | 1 −3 2 2 3 −4 2 2 = 2 eδ (|D u| − 8 eδ |∇|∇u| | + RicM (∇u, ∇u)). ∞ We now consider ψ ∈ Cc (M \ {p}) and then take f = eδψ in the stability inequality (1) yielding 2 2 2 2 2 1 2 2 2 2 |AM | eδ ψ ≤ |∇eδ| f + 2 ∇eδ , ∇ψ + eδ |∇ψ| ZM ZM 2 2 1 2 2 2 2 = |∇eδ| f − 2 ψ ∆eδ + eδ |∇ψ| ZM 2 2 2 = −eδψ ∆eδ + eδ|∇ψ| ZM 1 −2 2 2 3 −4 2 2 2 2 2 = − 2 eδ (|D u| − 8 eδ |∇|∇u| | + RicM (∇u, ∇u))ψ + eδ |∇ψ| . ZM Rearranging this yields 2 2 1 −2 2 2 3 −4 2 2 2 2 2 (|AM | eδ + 2 eδ (|D u| − 8 eδ |∇|∇u| | + RicM (∇u, ∇u)))ψ ≤ eδ |∇ψ| . ZM ZM 2 Note that RicM ≥ −|AM | by the Gauss equations. Furthermore, the improved Kato inequality yields 3 2 2 2 2 2 4 2 2 8 |∇|∇u| | ≤ |∇u| |D u| ≤ eδ |D u| . From this, we find that 2 2 1 −2 2 2 3 −4 2 2 |AM | eδ + 2 eδ (|D u| − 8 eδ |∇|∇u| | + RicM (∇u, ∇u)) ≥ 0. Let B be an open subset of (0, ∞) containing all singular values of u, and A = (0, ∞)\B. We thus find 2 2 1 −2 2 2 3 −4 2 2 2 (|AM | eδ + 2 eδ (|D u| − 8 eδ |∇|∇u| | + RicM (∇u, ∇u)))ψ A Z 2 2 2 2 ≤ eδ |∇ψ| + eδ |∇ψ| A B Z Z We can send δ → 0 and apply Fatou’s lemma to find 2 1 −1 2 2 3 2 2 (|AM | |∇u| + 2 |∇u| (|D u| − 2 |∇|∇u|| + RicM (∇u, ∇u)))ψ −1 A Zu ( ) ≤ |∇u||∇ψ|2 + |∇u||∇ψ|2 −1 A −1 B Zu ( ) Zu ( ) By the co-area formula, we find 2 1 −2 2 2 3 2 2 (|AM | + 2 |∇u| (|D u| − 2 |∇|∇u|| + RicM (∇u, ∇u)))ψ ds A Z ZΣs ≤ |∇ψ|2 ds + |∇u||∇ψ|2. A −1 B Z ZΣs Zu ( ) STABLE MINIMAL HYPERSURFACES IN R4 11 ∞ For ϕ ∈ Cc ((0, ∞)), take ψ = ϕ(u) to yield 2 2 1 −2 2 2 3 2 ϕ(s) |AM | + 2 |∇u| (|D u| − 2 |∇|∇u|| + RicM (∇u, ∇u)) ds A Z ZΣs ≤ ϕ′(s)2 |∇u|2 ds + ϕ′(u)2|∇u|3 A −1 B Z ZΣs Zu ( ) A ∇u If s ∈ , note that ν = |∇u| is a unit normal to Σs. We now follow the ideas used in [47, Theorem 1]. The traced Gauss equations yield 2 2 2 RicM (ν,ν)= RM − 2KΣs − |AΣs | + HΣs . 3 4 2 Similarly, using the Gauss equations for M → R we have RM = −|AM | . The scalar second fundamental form of Σs satisfies −1 2 AΣs = |∇u| D u|Σs so 2 2 2 2 2 2 2 |∇u| |AΣs | = |D u| − 2|∇|∇u|| + D u(ν,ν) and (because u is harmonic) 2 2 2 2 |∇u| HΣs = D u(ν,ν) . Thus, 1 2 2 2 1 2 2 2 RicM (∇u, ∇u)= − 2 |∇u| |AM | − |∇u| KΣs − 2 |D u| + |∇|∇u|| . along Σs. Thus, 2 3 2 1 −2 2 2 2 ϕ(s) 4 |AM | + 4 |∇u| (|D u| − |∇|∇u|| ) ds A Z ZΣs 2 1 ′ 2 2 ′ 2 3 ≤ ϕ(s) 2 KΣs ds + ϕ (s) |∇u| ds + ϕ (u) |∇u| A A −1 B Z ZΣs Z ZΣs Zu ( ) By (6) in Proposition 7, for s ∈ R,Σ is connected, so K ≤ 4π (by Gauss– s Σs Σs Bonnet). Using this, the Kato inequality, and the definition of F (s), A(s), we find R 8π 4 4 ϕ(s)2A(s) ds ≤ ϕ(s)2 ds + ϕ′(s)2F (s) ds + ϕ′(u)2|∇u|3 A 3 A 3 A 3 −1 B Z Z Z Zu ( ) Since ϕ′(u)2|∇u|3 is uniformly bounded, we can send |B ∩ supp ϕ|→ 0 and conclude ∞ 8π ∞ 4 ∞ ϕ(s)2A(s) ds ≤ ϕ(s)2 ds + ϕ′(s)2F (s) ds. 3 3 Z0 Z0 Z0 A standard approximation argument for ϕ completes the proof. Remark 12. By using the improved Kato inequality it is easy to generalize the previous argument to prove that ∞ ∞ 2 3 2 1 2 ′ 2 ϕ(s) 4 |AM | + 12 |AΣs | − KΣs ds ≤ ϕ (s) F (s) ds, Z0 ZΣs Z0 where AΣs is the second fundamental form of Σs in M. However, we will not need this expression in the sequel. 12 OTIS CHODOSH AND CHAO LI We thus see that the behavior of F (s) near 0 and ∞ determines the potential test functions ϕ that can be used in the inequality from Proposition 11. The behavior of F (t) as t → ∞ is independent of the geometry of M (F (t) behaves like the Green’s function on R3). However, the behavior as t → 0 is of crucial importance to our argument. We begin with an a priori bound that will be improved in the sequel. Lemma 13. It holds that F (t)= O(t) as t → 0 and F (t)=(1+ o(1))4πt2 as t →∞. Proof. The bound as t →∞ follows in a straightforward manner from the asymptotics in (3) in Proposition 7. For the other assertion, by integrating ∆u = 0 over {t ≤ u ≤ τ} (where t,τ ∈ R) we find |∇u| = |∇u|. ZΣt ZΣτ By Lemma 9, t 7→ |∇u| is constant. Because M has uniformly bounded Ricci Σt curvature, we can apply the Harnack inequality (Proposition 22) to obtain R F (t)= |∇u|2 ≤ Ct |∇u|. ZΣt ZΣt (for t bounded away from ∞). This proves the assertion. Corollary 14. We have t 1 4 1 lim sup A(s) ds + t2 s−2A(s) ds ≤ O(t)+ t2s−4F (s) ds 3 ℓց0 Zℓ Zt Zt as t → 0. Proof. For ε ∈ (0, 1) and ℓ 4. An extension of Munteanu–Wang’s monotonicity formula Munteanu–Wang have recently established [31] a sharp monotonicity formula for the quantity F (s) defined via a Green’s function on a non-parabolic three-manifold with non-negative scalar curvature. In this section we refine this estimate in two ways to apply in the present situation. First of all, the minimal hypersurface does not have non- negative scalar curvature so we must keep track of the resulting defect term. Second, we have to regularize their argument to account for our relatively weak a priori bounds for F (s) as s → 0. We continue to consider M 3 → R4 a complete, connected, simply connected, two- sided stable minimal immersion with uniformly bounded curvature. For p ∈ M fixed, ∞ consider the Greens’ function u ∈ Cloc(M \ {p}) constructed in Proposition 7. Recall the definition of F (s), A(s) in (5) and (6). By Lemma 9, F (s), A(s) are continuous on (0, ∞). Furthermore, by Lemma 10, F (s) is locally absolutely continuous. Finally, if R is the open and dense (by Lemma 8) set of regular values of u, we see that F (s) is differentiable at all s ∈ R. We consider α 7→ λ(α) where 1 λ(α) := α + 1 − (α + 1)(1 − 3 α). (7) Note that λ(α) ∈ R for α ∈ [−1, 3]. q 3 Lemma 15. There is α0 ∈ (1, 2) so that if α ∈ (α0, 2) then α − 2 λ(α) + 1 > 0. Proof. Taylor expanding around α = 2 yields 3 2 α − 2 λ(α)+1= −(α − 2) + O((α − 2) ). This proves the assertion. Proposition 16 (cf. [31, Theorem 3.1]). There is C > 0 so that for t ∈ (0, 1), it holds that 1 t 1 1 F (t) ≤ Ct3 + 4πt2 + t lim inf A(s) ds + t3 s−2A(s) ds. 4 ℓց0 4 Zℓ Zt ∇u Proof. We begin by considering t ∈ R. Set ν = |∇u| (note that ν is inwards pointing for −1 the set Ωs = {u ≥ s}). The family t 7→ Σt has normal velocity |∇u| ν. Furthermore, the mean curvature of Σs satisfies H = −|∇u|−1 h∇|∇u|,νi . With this convention, we have F ′(t)= |∇u|−1 ∇|∇u|2,ν + |∇u|−1H|∇u|2 = h∇|∇u|,νi (8) ZΣt ZΣt Fix α ∈ (α0, 2) as in Lemma 15 and write λ = λ(α) (defined in (7)). Note that t−αF ′(t)+ αt−α−1F (t)= u−α h∇|∇u|,νi + αu−α−1|∇u|2 ZΣt = u−α h∇|∇u|,νi−|∇u| ∇u−α,ν . ZΣt 14 OTIS CHODOSH AND CHAO LI Set 2 1 jδ = (|∇u| + δ) 2 so that −α ′ −α−1 −α −α t F (t)+ αt F (t) = lim u h∇jδ,νi− jδ ∇u ,ν . δց0 ZΣt Now, if 0 −α −α −α −α u h∇jδ,νi− jδ ∇u ,ν − u h∇jδ,νi− jδ ∇u ,ν ZΣt ZΣτ −α −α = − u ∆jδ − jδ∆u . ZΩt,τ (Recall that ν is inwards pointing for Ωs.) Because u is harmonic ∆u−α = α(α + 1)u−α−2|∇u|2. Furthermore, the Bochner formula yields −1 2 2 1 −2 2 2 −1 ∆jδ = jδ (|D u| − 4 jδ |∇|∇u| | )+ jδ RicM (∇u, ∇u). Thus, we find −α −α −α −α u h∇jδ,νi− jδ ∇u ,ν − u h∇jδ,νi− jδ ∇u ,ν ZΣt ZΣτ −α −1 2 2 1 −2 2 2 −α −1 = − u jδ (|D u| − 4 jδ |∇|∇u| | )+ u jδ RicM (∇u, ∇u) ZΩt,τ −α−2 2 + α(α + 1)u jδ|∇u| . ZΩt,τ Note that the Kato inequality yields 2 2 1 −2 2 2 |D u| − 4 jδ |∇|∇u| | ≥ 0. Let B be an open subset of (0, ∞) containing all singular values of u, and A = (0, ∞)\B. We thus find −α −α −α −α u h∇jδ,νi− jδ ∇u ,ν − u h∇jδ,νi− jδ ∇u ,ν ZΣt ZΣτ −α −1 −1 2 2 1 −2 2 2 ≤− u |∇u| jδ (|D u| − 4 jδ |∇|∇u| | ) ds A Z(t,τ)∩ ZΣs −α −1 −1 − u |∇u| jδ RicM (∇u, ∇u) ds A Z(t,τ)∩ ZΣs −α−2 + α(α + 1)u jδ|∇u| ds A Z(t,τ)∩ ZΣs −α−2 2 −α −1 + α(α + 1)u jδ|∇u| − u jδ RicM (∇u, ∇u) −1 B ZΩt,τ ∩u ( ) STABLE MINIMAL HYPERSURFACES IN R4 15 −α−2 2 −α −1 ∞ Because α(α+1)u jδ|∇u| −u jδ RicM (∇u, ∇u) is uniformly bounded in L (Ωt,τ ) as δ → 0, we can send δ → 0 and then |B| → 0 to find (t−αF ′(t)+ αt−α−1F (t)) − (τ −αF ′(τ)+ ατ −α−1F (τ)) τ ≤− u−α|∇u|−2(|D2u|2 − |∇|∇u||2) ds t Σs Z τ Z −α −2 − u |∇u| RicM (∇u, ∇u) ds t Σs Z τ Z + α(α + 1)u−α−2|∇u|2 ds. Zt ZΣs Using Stern’s rearrangement [47] of the Bochner terms as in the proof of Proposition 11 we can write (along Σs, s ∈ R) −2 1 2 −2 2 1 −2 2 2 |∇u| RicM (∇u, ∇u)= − 2 |AM | − KΣs + |∇u| |∇|∇u|| − 2 |∇u| |D u| so (also using that u is constant along its level sets) t−αF ′(t)+ αt−α−1F (t) − τ −αF ′(τ)+ ατ −α−1F (τ) 1 τ τ ≤ s−α |A |2 + s−α K ds 2 M Σs Zt ZΣs Zt ZΣs 1 τ τ − s−α |∇u|−2|D2u|2 ds + α(α + 1)s−α−2F (s)ds. 2 Zt ZΣs Zt By Gauss–Bonnet and Proposition 7, τ 1 s−α K ds ≤ 4π(t1−α − τ 1−α). Σs α − 1 Zt ZΣs Furthermore, using the improved Kato inequalty, as well as the Cauchy–Schwarz and H¨older inequalities on (8), we find −2 2 2 3 −2 2 |∇u| |D u| ≥ 2 |∇u| |∇|∇u|| ZΣs ZΣs 3 −2 2 ≥ 2 |∇u| h∇|∇u|,νi ZΣs 3 −1 ′ 2 ≥ 2 F (s) F (s) for s ∈ R. Putting this together we find t−αF ′(t)+ αt−α−1F (t) − τ −αF ′(τ)+ ατ −α−1F (τ) 1 τ 1 ≤ s−αA(s) ds + 4π(t1−α − τ 1−α) 2 t α − 1 Z τ 3 −α −1 ′ 2 −α−2 + (− 4 s F (s) F (s) + α(α + 1)s F (s))ds Zt Cauchy–Schwarz yields 2s−1F ′(s)F (s) ≤ λ−1F ′(s)2 + λs−2F (s)2, 16 OTIS CHODOSH AND CHAO LI (where λ = λ(α) as defined in (7)). Rearranging, we find 3 −α −1 ′ 2 3 −α−1 ′ 3 2 −α−2 − 4 s F (s) F (s) ≤− 2 λτ F (s)+ 4 λ s F (s). Using this above we find t−αF ′(t)+ αt−α−1F (t) − τ −αF ′(τ)+ ατ −α−1F (τ) 1 τ 1 ≤ s−αA(s) ds + 4π(t1−α − τ 1−α) 2 t α − 1 Z τ 3 −α−1 ′ 3 2 −α−2 + (− 2 λs F (s) + ( 4 λ + α(α + 1))s F (s)) ds Zt 1 τ 1 = s−αA(s) ds + 4π(t1−α − τ 1−α) 2 t α − 1 Z τ 3 2 3 −α−2 + ( 4 λ − 2 λ(α +1)+ α(α + 1))s F (s) ds Zt 3 −α−1 3 −α−1 − 2 λτ F (τ)+ 2 λt F (t), where we integrated by parts in the last step. Observe that 3 2 3 4 λ − 2 λ(α +1)+ α(α + 1) = 0 by (7), so we can rearrange this to read −α ′ 3 −α−1 −α ′ 3 −α−1 t F (t) + (α − 2 λ)t F (t) − τ F (τ) + (α − 2 λ)τ F (τ) 1 τ 1 ≤ s−αA(s) ds + 4π(t1−α − τ 1−α) 2 α − 1 Zt 1 τ 1 ≤ s−αA(s) ds + 4πt1−α. 2 α − 1 Zt We thus find ′ τ α 3 λ α 3 λ 1 α 3 λ 1 α 3 λ t − 2 F (t) ≤ Ct2 − 2 + 4πt − 2 +1 + t2 − 2 s−αA(s) ds α − 1 2 Zt where C = C(τ) is bounded uniformly for α ∈ (α0, 2). By Lemma 10 we can integrate this on (ℓ,t) for ℓ,t ∈ R, ℓ 3 3 3 α λ α λ C α λ − 2 − 2 2 − 2 +1 t F (t) ≤ ℓ F (ℓ)+ 3 t 2α − 2 λ + 1 3 1 α λ − 2 +2 + 3 4πt (α − 1)(α − 2 λ + 2) t τ 1 α 3 λ + σ2 − 2 s−αA(s) dsdσ 2 Zℓ Zσ STABLE MINIMAL HYPERSURFACES IN R4 17 (where we have dropped several negative terms evaluated at ℓ, cf. Lemma 15). We apply Fubini’s theorem to write t τ 3 α λ σ2 − 2 s−αA(s) dsdσ Zℓ Zσ t s 3 τ t 3 α λ α λ = σ2 − 2 s−αA(s) dσds + σ2 − 2 s−αA(s) dσds Zℓ Zℓ Zt Zℓ t 3 3 τ 1 α λ 1 α λ ≤ s − 2 +1A(s) ds + t2 − 2 +1 s−αA(s) ds. 2α − 3 λ + 1 2α − 3 λ + 1 2 Zℓ 2 Zt α 3 λ By Lemma 15, s − 2 +1 ≤ 1 for s ∈ (0, 1). Because t< 1 we can thus estimate t τ 3 α λ σ2 − 2 s−αA(s) dsdσ Zℓ Zσ t 3 τ 1 1 α λ ≤ A(s) ds + t2 − 2 +1 s−αA(s) ds. 2α − 3 λ + 1 2α − 3 λ + 1 2 Zℓ 2 Zt On the other hand, by Lemmas 13 and 15 we have that 3 3 α λ α λ ℓ − 2 F (ℓ)= O(ℓ − 2 +1)= o(1) as ℓ → 0. Thus, we can pass to the limit as ℓ ց 0 to obtain 3 3 3 α λ C α λ 1 α λ − 2 2 − 2 +1 − 2 +2 t F (t) ≤ 3 t + 3 4πt 2α − 2 λ + 1 (α − 1)(α − 2 λ + 2) 1 t + lim inf A(s) ds 2(2α − 3 λ + 1) ℓց0 2 Zℓ 3 τ 1 α λ + t2 − 2 +1 s−αA(s) ds. 2(2α − 3 λ + 1) 2 Zt Because α 7→ λ(α) is continuous at α = 2 and λ(2) = 2 we can then send α ր 2 to find 1 t 1 τ t−1F (t) ≤ Ct2 + 4πt + lim inf A(s) ds + t2 s−2A(s) ds 4 ℓց0 4 Zℓ Zt Because τ ≤ 1, this yields the assertion. Remark 17. Using the method of proof of Proposition 16 it should be possible to weaken the hypothesis lim infx→∞ Ric ≥ 0 in [31, Corollary 3.2] to lim infx→∞ Ric > −∞. Finally, we are able to obtain the following sharp decay estimate. Corollary 18. We have F (t)= O(t2) as t → 0. Proof. Combining Proposition 16 with Corollary 14 we obtain 1 1 F (t) ≤ O(t2)+ t3 s−4F (s) ds (9) 3 Zt 18 OTIS CHODOSH AND CHAO LI −2 as t → 0. Assume for contradiction that lim suptց0 F (t)t = ∞. If this held, then we could choose {tj}j∈N ⊂ (0, 1) so that −2 −2 F (tj)tj = max F (s)s →∞. [tj ,1] Using (9) at t = tj we find 1 1 F (t ) ≤ O(t2)+ t3 s−2(s−2F (s)) ds j j 3 j Ztj 1 1 ≤ O(t2)+ t F (t ) s−2 ds j 3 j j Ztj 1 = O(t2)+ F (t )(1 − t ). j 3 j j Rearranging this we find 1 2 (1 − 3 (1 − o(1)))F (tj ) ≤ O(tj ). This is a contradiction, completing the proof. 5. Proof of Theorems 1, 2, and 3 Theorem 1 follows from Theorem 2 and the fact that there are no compact minimal surfaces in Rn+1. To prove Theorems 2 and 3, we briefly recall a standard point-picking argument (cf. [50, Lecture 3]) as follows: if Theorem 2 (or Theorem 3) was not true, then we could find a sequence of two-sided, stable minimal immersed hypersurfaces Mi 4 4 in R (or in (N , g)) and pi ∈ Mi such that |AMi (pi)|dMi (pi, ∂Mi)= Ri →∞. Here the distance is intrinsic on Mi. By considering an appropriate subset of Mi we can assume that Mi is compact and smooth up to its boundary. This allows us to assume that pi maximizes |Ai(x)|dM (x,∂Mi). By translating and rescaling (we still denote the surfaces by Mi), we can ensure that pi = 0 and |Ai(0)| = 1. Then, for any r < Ri and x ∈ Mi with dMi (0,x) ≤ r we find Ri Ri |AMi (x)|≤ ≤ , dMi (x,∂Mi) Ri − r and thus for each r> 0, Ri sup |AMi (x)|≤ → 1. Ri − r dMi (x,0)≤r Therefore Mi subsequentially converges smoothly to a complete, two-sided, stable min- 3 4 imal immersion M → R with |AM (0)| = 1 and |AM (x)|≤ 1 for all x ∈ M (this last condition was not assumed a priori in the statement of Theorem 1). To show that such an immersion does not exist, we first pass to the universal cover to arrange that assume that M is simply connected (two-sided stability passes to any STABLE MINIMAL HYPERSURFACES IN R4 19 ∞ covering space by [21, Theorem 1]). We can construct the Green’s function u ∈ Cloc(M \ {p}) as in Proposition 7 and conclude that F (t)= |∇u|2 ZΣt satisfies F (t) ≤ Ct2 for all t ∈ (0, ∞) by Lemma 13 and Corollary 18. Consider f = ϕ◦u 0,1 for ϕ ∈ Cc ((0, ∞)) in Proposition 25 and apply the co-area formula to write 3 3 3 |AM | ϕ(u) ≤ C |∇(ϕ ◦ u)| ZM ZM = C ϕ′(u)3|∇u|3 M Z ∞ = C ϕ′(s)3 |∇u|2 ds 0 Σs Z ∞ Z = C ϕ′(s)3F (s) ds 0 Z ∞ ≤ C ϕ′(s)3s2 ds. Z0 For ρ ≫ 0 choose 0 t ∈ [0, ρ−2) log t −2 −1 2+ log ρ t ∈ [ρ , ρ ) ϕ(t)= 1 t ∈ [ρ−1, ρ) log t 2 2 − log ρ t ∈ [ρ, ρ ) 2 0 t ∈ [ρ , ∞). We find 3 |AM | {ρ−1≤u≤ρ} Z − ρ 1 2 ρ2 2 s s −2 ≤ C 3 3 ds + C 3 3 ds = O(| log ρ| ). −2 s | log ρ| s | log ρ| Zρ Zρ Letting ρ →∞, we find that AM ≡ 0. This completes the proof. 6. Proof of Theorem 4 3 4 3 By [49, §3], a minimal immersion M → R with finite total curvature M |AM | < ∞ has finite index. Thus, it suffices to prove that finite index implies finite total curvature. Consider a complete, two-sided, minimal immersion M 3 → R4R with finite index. By Proposition 21, M has k < ∞ ends. Note that M is oriented and has bounded curvature. Lemma 19. Consider an exhaustion Ω1 ⊂ Ω2 ⊂···⊂ M by pre-compact regions with smooth boundary so that each component of M \Ωi is unbounded. Then for i sufficiently large, ∂Ωi has k components. 20 OTIS CHODOSH AND CHAO LI Proof. Discarding finitely many terms, we can assume M \Ωi has k components for all i. If the assertion fails, we can pass to a subsequence so that Ωi+1 \ Ωi has k components and ∂Ωi has at least two components in a fixed end E. This yields a sequence of surfaces Σi ⊂ ∂Ωi that form a linearly independent set in H2(M). By Poincar´eduality, 1 this implies that Hc (M; R) is infinite. By [9, Proposition 2.11] and Proposition 23, this further implies that the first L2-Betti number is infinite, contradicting Proposition 21. Consider an end E ⊂ M. We can assume that E has smooth boundary. Using the arguments in Proposition 7, we can construct a harmonic function u ∈ C∞(E) with finite Dirichlet energy so that u =1 on ∂E and u → 0 at infinity. By Lemma 19, we can choose τ0 ∈ (0, 1) a regular value of u, so that for any other regular value t ∈ (0,τ0), −1 Σt := u (t) is connected and {u>τ0} is stable. The proof of Corollary 18 carries over to this situation to show that F (t)= |∇u|2 ZΣt satisfies F (t)= O(t2). We proceed essentially as in the proof of Theorems 1 and 2, and plug the test function ϕ(u) into Proposition 25, where 0 t ∈ [0, ρ−2) 2+ log t t ∈ [ρ−2, ρ−1) log ρ ϕ(t)= τ 1 t ∈ [ρ−1, 0 ) 2 2 − 2t t ∈ [ τ0 ,τ ]. τ0 2 0 Sending ρ →∞, we conclude that 3 |AM | < ∞. ZE Since there are finitely many ends (Proposition 21), this yields 3 |AM | < ∞, ZM as desired. Appendix A. Ends of stable (finite index) minimal hypersurfaces Proposition 20 ([8]). For n> 2, if M n → Rn+1 is a complete, connected, two-sided, stable minimal immersion then M has only one end. Proposition 21 ([27]). For n> 2, if M n → Rn+1 is a complete, connected, two-sided, stable minimal immersion with finite index, then M has finitely many ends. Moreover, the space of L2-harmonic 1-forms is finite. STABLE MINIMAL HYPERSURFACES IN R4 21 Appendix B. Harnack and Sobolev inequalities Proposition 22 ([51], cf. [38, Theorem I.3.1]). Suppose that (M n, g) is a complete 2 Riemannian manifold and u is a positive harmonic function on Br(x). If Ric ≥ −K on Br(x) then |∇u|(x) ≤ C(r−1 + K)u(x) for C = C(n). Proposition 23 ([29]). For n > 2, suppose that M n → Rn+1 is a complete minimal 0,1 immersion. Then for any w ∈ Cc (M), it holds that n−2 2n n w n−2 ≤ C |∇w|2 ZM ZM for C = C(n). Appendix C. Level sets of harmonic functions Proposition 24 ([24, 28, 10]). Fix a compact subset K of a Riemannian n-manifold (M n, g), if ∆u = 0 is a harmonic function on (M, g) then n−1 n−1 H (Bρ(x) ∩ {u = s}) ≤ Cρ for any s ∈ R, x ∈ K, ρ ≤ ρ0 = ρ0(M, g, K, u) where C = C(M, g, K, u). Furthermore dimHau({u = s, |∇u| = 0}) ≤ n − 2 for any s ∈ R. Appendix D. The Schoen–Simon–Yau L3 estimate Schoen–Simon–Yau have obtained Lp estimates for the second fundamental form of a stable minimal hypersurface M n → Rn+1 in [37]. The following L3 estimate follows from their arguments in a straightforward manner but we could not find it in the literature (the estimate from [37] is only stated for p ≥ 4) so we include the proof. Proposition 25. For n < 8, if M n → Rn+1 a complete, connected, two-sided, stable minimal immersion then there is C = C(n) so that 3 3 3 |AM | f ≤ C |∇f| . ZM ZM 0,1 for any f ∈ Cc (M). Proof. Set 2 1 aδ = (|AM | + δ) 4 . ∞ For f ∈ Cc (M) we can take aδf in the stability inequality to find 2 2 2 |AM | aδ f ZM 2 2 2 2 2 ≤ |∇aδ| f + f ∇aδ, ∇f + aδ|∇f| ZM 1 −6 2 2 2 1 −2 2 2 2 = 16 aδ |∇|AM | | f + 2 faδ ∇|AM | , ∇f + aδ |∇f| (10) ZM 22 OTIS CHODOSH AND CHAO LI Multiplying Simons identity [45] 2 1 2 4 |∇AM | = 2 ∆|AM | + |AM | −2 2 by aδ f and integrating by parts we find −2 2 2 aδ |∇AM | f ZM 1 −2 2 2 −2 4 2 = 2 aδ f ∆|AM | + aδ |AM | f ZM 1 −6 2 2 2 −2 2 −2 4 2 = 4 aδ |∇|AM | | f − aδ f ∇f, ∇|AM | + aδ |AM | f ZM so −2 2 2 1 −4 2 2 2 aδ (|∇AM | f − 4 aδ |∇|AM | | f ) ZM −2 2 −2 4 2 = −aδ f ∇f, ∇|AM | + aδ |AM | f . ZM Note that the improved Kato inequality reads 1 2 2 2 2 4 2 4 (1 + n )|∇|AM | | ≤ |AM | |∇AM | ≤ aδ |∇AM | so we can rearrange this into 1 −6 2 2 −2 2 −2 4 2 2n aδ |∇|AM | |f ≤ −aδ f ∇f, ∇|AM | + aδ |AM | f . ZM ZM −2 −1 Because aδ ≤ |AM | we find 1 −6 2 2 −2 2 3 2 2n aδ |∇|AM | |f ≤ −aδ f ∇f, ∇|AM | + |AM | f , ZM ZM so we can combine this with stability to write 8−n −6 2 2 1 −2 2 2 2 16n aδ |∇|AM | |f ≤ − 2 aδ f ∇f, ∇|AM | + aδ|∇f| ZM ZM Because n< 8 we can use Cauchy–Schwartz to find C = C(n) so that −6 2 2 2 2 aδ |∇|AM | |f ≤ C aδ |∇f| (11) ZM ZM On the other hand, we can use Cauchy–Schwartz on (10) to find 2 2 2 17 −6 2 2 2 17 2 2 |AM | aδ f ≤ 16 aδ |∇|AM | | f + 16 aδ |∇f| . (12) ZM ZM Combining (11) and (12) and taking the limit as δ → 0 (using the dominated conver- gence theorem) we find 3 2 2 |AM | f ≤ C |AM ||∇f| . ZM ZM 0,1 A standard argument shows that this holds for f ∈ Cc (M). We can then replace f 3 by f 2 and using H¨older’s inequality to conclude the proof. STABLE MINIMAL HYPERSURFACES IN R4 23 References [1] V. Agostiniani, L. Mazzieri, and F. Oronzio, A Green’s function proof of the positive mass theorem, https://arxiv.org/abs/2108.08402 (2021). 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