arXiv:2107.12784v1 [math.DG] 27 Jul 2021 aiod C2] n yeblcpstv asterm[BHK+21]. of theorem torus mass mapping positive and hyperbolic cubes and Gromov’s develop dimension [CK21], HKK20], Later manifolds three [BKKS19, for theorem version. conjecture mass boundary rigidity positive Neumann of a proofs for simplified [BS19] geome curvature See scalar in results ifolds. important some of proofs simplify odto.W aeasmls xml:i he aiod( manifold three a if example: h simplest a take We condition. equation Jang a also µ See [ACG08]). (see function ex aspect under c mass manifolds of the hyperbolic study on asymptotically a in via surface established 2 be surface curvature could trapped theorem outer mass marginally positive stable hyperbolic and [SY81] equation Jang (0.1) a esilcasfiduigtesaiiycniinad[FS80]. and condition stability the using classified still be can hntetplg fastable a of topology the then oda h ekrcniinsmlrt 01.A nta aast( set data initial An (0.1). to similar condition weaker the deal to imninmnfl ( manifold Riemannian rpsto 0.1. Proposition proposition. analogous the establish 03 ∆ (0.3) h qaini etraint h pctm amncfunction, harmonic spacetime the to u perturbation a is equation The ie mohfunction smooth a Given e ihsot boundary smooth with set pctm amncfnto,addnt h pnsbe of subset open the denote and function, harmonic spacetime (0.2) P bbls h ei fastable a of merit The -bubbles. ETRAINO PCTM ALCA EQUATION LAPLACIAN SPACETIME OF PERTURBATION A ∈ ssaeiehroi if harmonic spacetime is tr = tr Se9 nrdcdalvlstmto fhroi asadh and maps harmonic of method set level a introduced [Ste19] Stern rmv[r1,Go1 ute eeaie iia ufc appro surface minimal generalized further Gro21] [Gro18, Gromov su minimal stable used theorems mass positive to approach first The egnrlz h saeie amncfnto praho [BKKS of approach function harmonic (spacetime) the generalize We C ∞ p ( h pctm esa HK0 sdfie obe to defined is [HKK20] Hessian spacetime the , Abstract. fsaeieLpaineuto na nta aast( set data initial an in equation Laplacian spacetime of rc ftesmerc2-tensor symmetric the of trace M uhta h clrcurvature scalar the that such ) esuyaperturbation a study We Let ,g M, ( ∇ ,g p g, U, ¯ h ∂U h 2 qipdwt netasmerc2-form symmetric extra an with equipped ) on u .W olw[K2,Pooiin32 ndti and detail in 3.2] Proposition [HKK20, follow We 0. = aigotadui normal unit outward having , = R µ M ) bbl fpecie encurvature mean prescribed of -bubble g ∆ ∇ u IOIN CHAI XIAOXIANG ea3dmninloine opc nta data initial compact oriented 3-dimensional a be esuyteslto oteequation the to solution the study we , + µ u 2 + bbl spsiiiyo ekrsaa curvature scalar weaker of possibility is -bubble u + p h P + 2 P and |∇ − |∇ p |∇ u 1 u 2 h | |∇ | sasot function. smooth a is u = = R | g u , h g h h |∇ | |∇ satisfies ∈ > u | u C , 0 | . 2 , ( M ,g p g, M, ) . ,g M, η ∂U where ) Let . disafunction a admits ) nwhich on EL1] The [EHLS16]. s r ftreman- three of try n h solution the and u P r assumption tra et include ments ro [Sak21]. proof : ntn mean onstant yeblc3- hyperbolic sthe is c ostable to ach fc [SY79], rface ,g p g, M, h U 9 HKK20] 19, p sdi to it used e ewrite We . n( in → |∇ dihedral M R u sa is ) 6 | 3 ea be 0 = g , ) 2 XIAOXIANG CHAI by ∂6=0U. If u¯ and u denote the maximum and minimum values of u and Σt are t-level sets, then

(∂η|∇u| + p(∇u, η))dA Z∂6=0U u¯ 1 |∇¯ 2u|2 > 2 (0.4) 2 + (µ + J(ν)+ h − 2hP +2hν, ∇hi) − K dAdt, Zu ZΣt 2 |∇u|  ∇u where ν = |∇u| , K is Gauss curvature of the level set and dA is the area element. Proof. Recall the Bochner formula 1 ∆|∇u|2 = |∇2u|2 + Ric(∇u, ∇u)+ h∇u, ∇∆ui. 2 1 2 For δ > 0 set φδ = (|∇u| + δ) 2 , and use the Bochner formula to find ∆|∇u|2 |∇|∇u|2|2 ∆φδ = − 3 2φδ 4φδ 1 (0.5) > (|∇2u|2 − |∇|∇u||2 + Ric(∇u, ∇u)+ h∇u, ∇∆ui). φδ ∇u On a regular level set Σ, the unit normal is ν = |∇u| and the second fundamental ∇i∇j u form is given by Aij = |∇u| , where ∂i and ∂j are tangent to Σ. We then have |A|2 = |∇u|−2(|∇2u|2 − 2|∇|∇u||2 + [∇2u(ν,ν)]2), and the mean curvature satisfies 2 (0.6) |∇u|H = ∆u − ∇ν u Furthermore by taking two traces of the Gauss equations 2 2 2Ric(ν,ν)= Rg − RΣ − |A| + H , where Rg is the scalar curvature of U and RΣ is the scalar curvature of the level set Σ. Combining these formulas with (0.5) produces 2 1 2 2 2 |∇u| 2 2 ∆φδ > |∇ u| − |∇|∇u|| + h∇u, ∇∆ui + (Rg − RΣ + H − |A| ) φδ  2 

1 2 2 2 2 2 = (|∇ u| + (Rg − RΣ)|∇u| +2h∇u, ∇∆ui + (∆u) − 2(∆u)∇ν u). 2φδ Let us now replace the Hessian with the spacetime Hessian via the relation (0.2), and utilize (0.3) to find

∆φδ

1 2 2 2 > (|∇¯ u − p|∇u|| + (Rg − RΣ)|∇u| +2h∇u, ∇(−P |∇u| + h|∇u|)i 2φδ 2 2 + (−P |∇u| + h|∇u|) − 2(−P |∇u| + h|∇u|)∇ν u). Noting that 1 h∇u, ∇|∇u|i = hν, ∇|∇u|2i = ui∇ u = |∇u|∇2 u, 2 iν ν A PERTURBATION OF SPACETIME LAPLACIAN EQUATION 3 and expanding |∇¯ 2u − p|∇u||2, we have that

∆φδ 1 ¯ 2 2 2 2 2 2 > (|∇ u| − 2hp, ∇ ui|∇u| − |p|g|∇u| + (Rg − RΣ)|∇u| 2φδ +2|∇u|h∇u, −∇P + ∇hi + (−P |∇u| + h|∇u|)2).

2 2 Regrouping using the energy density 2µ = Rg + P − |p|g,

∆φδ

1 2 2 2 2 > [|∇¯ u| − 2hp, ∇ ui|∇u| + (2µ − RΣ)|∇u| − 2|∇u|h∇u, ∇P i 2φδ (0.7) + |∇u|2(h2 − 2hP +2hν, ∇hi]. Consider an open set A⊂ [u, u¯] containing the critical values of u, and let B⊂ [u, u¯] denote the complementary closed set. Then integrate by parts to obtain

h∇φδ , vi = ∆φδ = ∆φδ + ∆φδ. Z∂U ZU Zu−1(A) Zu−1(B) According to Kato type inequality [HKK20, Lemma 3.1] and (0.5) there is a positive constant C0, depending only on Ricg, P − h and its first derivatives such that

∆φδ > −C0|∇u|. An application of the coarea formula to u : u−1(A) → A then produces

(0.8) − ∆φε 6 C0 |∇u| = C0 |Σt|dt, Zu−1(A) Zu−1(A) Zt∈A where |Σt| is the 2-dimensional Hausdorff measure of the t-level set Σt. Next, apply the coarea formula to u : u−1(B) →B together with (0.7) to obtain

∆φδ Zu−1(B) 1 |∇u| |∇¯ 2u|2 2 > 2 2 +2µ − RΣt − (hp, ∇ ui + h∇u, ∇P i) dAdt 2 Zt∈B ZΣt φδ  |∇u| |∇u|  1 1 + (h2 − 2hP +2hν, ∇hi)dAdt. 2 Zt∈B ZΣt φδ Combining with (0.8) and

∆φδ = ∆φδ − ∆φδ = ∂ηφδ − ∆φδ, Zu−1(B) ZU Zu−1(A) Z∂U Zu−1(A) we obtain

∂ηφδ + C0 |Σt|dt Z∂U Zt∈A 1 |∇u| |∇¯ 2u|2 2 > 2 2 +2µ − RΣt − (hp, ∇ ui + h∇u, ∇P i) dAdt 2 Zt∈B ZΣt φδ  |∇u| |∇u|  1 1 (0.9) + (h2 − 2hP +2hν, ∇hi)dAdt. 2 Zt∈B ZΣt φδ 4 XIAOXIANG CHAI

On the set u−1(B), we have that |∇u| is uniformly bounded from below. In addition, on ∂6=0U it holds that |∇u| ∂ηφδ = ∂η|∇u| → ∂η|∇u| as δ → 0. φδ Therefore, the limit δ → 0 may be taken in (0.9), resulting in the same bulk expression except that φε is replaced by |∇u|, and with the boundary integral taken over the restricted set. Furthermore, by Sard’s theorem (see [HKK20, Remark 3.3]) the measure |A| of A may be taken to be arbitrarily small. Since the map t 7→ |Σt| is integrable over [u, u¯] in light of the coarea formula, by then taking |A| → 0 we obtain Lastly integration by parts gives

2 ij i j hp, ∇ ui = p ∇ij u = p(∇u, η) − u ∇ pij , ZU ZU Z∂U ZU and recalling that J = divg(p − Pg) and RΣt =2KΣt , yields the desired result.  Now we discuss two special boundary conditions of the equation (0.3) namely Dirichlet and Neumann boundary conditions and its geometric implications on the boundary contribution

(∂η|∇u| + p(∇u,υ))dA Z∂6=0U in (0.4). We assume that u ∈ C2(U ∪ S) where S is a relatively open portion of ∂U. This is a valid assumption due to an existence theorem of [HKK20, Section 4]. We have the following: Lemma 0.2. Assume the solution u to (0.3) takes constant values on S, then

∂η|∇u| + p(∇u, η) = (−HS − trS p + h)|∇u|. Proof. Since u is constant on S, then by using the decomposition of the Laplacian ∆, 2 −P |∇u| + h|∇u| = ∆u = HSh∇u, ηi + ∇ u(η, η). The decomposition is already used in (0.6). Since u is constant on S, ∇u either point outward or inward of U. We calculate only the case when ∇u points outward, that is η = ∇u/|∇u|,

∂η|∇u| + p(∇u, η) 1 2 = |∇u| (∇ u)(∇u, η)+ p(∇u, η) =(∇2u)(η, η)+ p(η, η)|∇u|

= − HS |∇u|− P |∇u| + p(η, η)|∇u| + h|∇u|

=(−H − trS p + h)|∇u|. 

∂u Lemma 0.3. Suppose that ∂η =0 on S, then 1 ∂η|∇u| = |∇u| B(∇u, ∇u), where B is the second fundamental form of S in U. A PERTURBATION OF SPACETIME LAPLACIAN EQUATION 5

Proof. First,

∂η|∇u| 1 i j = |∇u| ηj ∇iu∇ ∇ u 1 i j i j i = |∇u| [∇iu∇ (ηj ∇ u) − ∇ u∇ u∇ ηj ] 1 = − |∇u| B(∇u, ∇u), we have used the boundary condition ∂u/∂η = 0. 

Note that we have not used u is a solution to (0.3). On a regular level set Σt, ∇u let e1 be a unit tangent vector of ∂Σt, then {η,ν = |∇u| ,e1} forms an orthonormal basis. We see that the geodesic curvature of ∂Σt in Σt is given by

κ∂Σt = h∇e1 η,e1i

∂u since ∂η = 0 implies that η is orthogonal to ∂Σt in Σt and points outward of Σt. So

B(ν,ν)= h∇ν η,νi

=(h∇ν η,νi + h∇e1 η,e1i) − h∇e1 η,e1i

=HS − κ∂Σt .

Therefore, we conclude the following.

Lemma 0.4. Assume the solution u to (0.3) takes constant values on S, at a point in S ∩ ∂Σt with Σt being a regular level set, then

∇u ∂η|∇u| + p(∇u, η) = (−HS + p( |∇u| ,η)+ κ∂Σt )|∇u|. So to study rigidity questions on initial data sets with boundary, it is natural to assume

µ + J(ν)+ h2 − 2hP +2|∇h| > 0, and the convexity condition on the boundary

HS > | trS p − h|, or

HS > p(η,e0) where e0 is any unit vector on S. With h = 0, the conditions on the boundary are termed boundary dominant energy conditions by [AdLM19] in their study of initial data sets with a noncompact boundary.

Acknowledgment Research of Xiaoxiang Chai is supported by KIAS Grants under the research code MG074402. I would also like to thank Tin Yau Tsang (UCI) and Sven Hirsch (Duke) for discussions on the level set technique of harmonic functions. 6 XIAOXIANG CHAI

References

[ACG08] Lars Andersson, Mingliang Cai, and Gregory J. Galloway. Rigidity and Positivity of Mass for Asymptotically Hyperbolic Manifolds. Annales Henri Poincar´e, 9(1):1–33, 2008. [AdLM19] Sergio Almaraz, Levi Lopes de Lima, and Luciano Mari. Spacetime positive mass theorems for initial data sets with noncompact boundary. 2019. [BHK+21] Hubert Bray, Sven Hirsch, Demetre Kazaras, Marcus Khuri, and Yiyue Zhang. Space- time Harmonic Functions and Applications to Mass. ArXiv:2102.11421 [gr-qc], feb 2021. [BKKS19] Hubert L. Bray, Demetre P. Kazaras, Marcus A. Khuri, and Daniel L. Stern. Harmonic Functions and The Mass of 3-Dimensional Asymptotically Flat Riemannian Manifolds. ArXiv:1911.06754 [gr-qc], 2019. [BS19] Hubert L. Bray and Daniel L. Stern. Scalar curvature and harmonic one-forms on three-manifolds with boundary. ArXiv:1911.06803 [math], 2019. [CK21] Xiaoxiang Chai and Inkang Kim. Scalar curvature, mean curvature and harmonic maps to the circle. ArXiv:2103.09737 [gr-qc], 2021. [EHLS16] Michael Eichmair, Lan-Hsuan Huang, Dan A. Lee, and Richard Schoen. The space- time positive mass theorem in dimensions less than eight. J. Eur. Math. Soc. (JEMS), 18(1):83–121, 2016. [FS80] Doris Fischer-Colbrie and Richard Schoen. The structure of complete sta- ble minimal surfaces in 3-manifolds of non-negative scalar curvature. Communications on Pure and Applied Mathematics, 33(2):199–211, 1980. [Gro18] Misha Gromov. Metric inequalities with scalar curvature. Geom. Funct. Anal., 28(3):645–726, 2018. [Gro21] Misha Gromov. Four Lectures on Scalar Curvature. ArXiv:1908.10612 [math], 2021. [HKK20] Sven Hirsch, Demetre Kazaras, and Marcus Khuri. Spacetime Harmonic Functions and the Mass of 3-Dimensional Asymptotically Flat Initial Data for the Einstein Equations. Journal of Differential Geometry, to appear, 2020. [Sak21] Anna Sakovich. The Jang Equation and the Positive Mass Theorem in the Asymptot- ically Hyperbolic Setting. Communications in Mathematical Physics, apr 2021. [Ste19] Daniel Stern. Scalar curvature and harmonic maps to S1. ArXiv:1908.09754 [math], 2019. [SY79] Richard Schoen and Shing Tung Yau. On the proof of the positive mass conjecture in general relativity. Comm. Math. Phys., 65(1):45–76, 1979. [SY81] Richard Schoen and Shing Tung Yau. Proof of the positive mass theorem. ii. Communications in Mathematical Physics, 79(2):231–260, 1981.

Korea Institute for Advanced Study, Seoul 02455, South Korea Email address: [email protected]