A PERTURBATION OF SPACETIME LAPLACIAN EQUATION XIAOXIANG CHAI Abstract. We study a perturbation ∆u + P |∇u| = h|∇u|, of spacetime Laplacian equation in an initial data set (M,g,p) where P is the trace of the symmetric 2-tensor p and h is a smooth function. Stern [Ste19] introduced a level set method of harmonic maps and he used it to simplify proofs of some important results in scalar curvature geometry of three man- ifolds. See [BS19] for a Neumann boundary version. Later developments include simplified proofs of positive mass theorem [BKKS19, HKK20], Gromov’s dihedral rigidity conjecture for three dimension cubes and mapping torus of hyperbolic 3- manifolds [CK21], and hyperbolic positive mass theorem [BHK+21]. The first approach to positive mass theorems used stable minimal surface [SY79], Jang equation [SY81] and stable marginally outer trapped surfaces [EHLS16]. The hyperbolic positive mass theorem could be established via a study of constant mean curvature 2 surface in asymptotically hyperbolic manifolds under extra assumption on the mass aspect function (see [ACG08]). See also a Jang equation proof [Sak21]. Gromov [Gro18, Gro21] further generalized minimal surface approach to stable µ-bubbles. The merit of a stable µ-bubble is possibility of weaker scalar curvature condition. We take a simplest example: if a three manifold (M,g) admits a function ∞ h ∈ C (M) such that the scalar curvature Rg satisfies 2 g (0.1) Rg + h − 2|∇ h| > 0, then the topology of a stable µ-bubble of prescribed mean curvature h in (M 3,g) can be still classified using the stability condition and [FS80]. We generalize the (spacetime) harmonic function approach of [BKKS19, HKK20] to deal the weaker condition similar to (0.1). An initial data set (M,g,p) is a Riemannian manifold (M,g) equipped with an extra symmetric 2-form p. We write arXiv:2107.12784v1 [math.DG] 27 Jul 2021 P = tr p, the spacetime Hessian [HKK20] is defined to be (0.2) ∇¯ 2u = ∇2u + p|∇u|,u ∈ C2(M). Given a smooth function h on M, we study the solution to the equation (0.3) ∆u + P |∇u| = h|∇u|. The equation is a perturbation to the spacetime harmonic function, and the solution u is spacetime harmonic if h = 0. We follow [HKK20, Proposition 3.2] in detail and establish the analogous proposition. Proposition 0.1. Let (U,g,p) be a 3-dimensional oriented compact initial data set with smooth boundary ∂U, having outward unit normal η. Let u : U → R be a spacetime harmonic function, and denote the open subset of ∂U on which |∇u| 6=0 1 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.