Parabolic constructions

Parabolic constructions of asymptotically flat 3-metrics of Prescribed scalar curvature

Chen-Yun Lin Department of Mathematics 196 Auditorium Road, Unit 3009 Storrs, CT 06269-3009, USA E-mail address: chen-yun.lin@uconn.edu
Abstract.

In 1993, Bartnik [3] introduced a quasi-spherical construction of metrics of prescribed scalar curvature on 3-manifolds. Under quasi-spherical ansatz, the problem is converted into the initial value problem for a semi-linear parabolic equation of the lapse function. The original ansatz of Bartnik started with a background foliation with round metrics on the 2-sphere leaves. This has been generalized by several authors [10, 14, 12] under various assumptions on the background foliation. In this article, we consider background foliations given by conformal round metrics, and by the Ricci flow on 2-spheres. We discuss conditions on the scalar curvature function and on the foliation that guarantee the solvability of the parabolic equation, and thus the existence of asymptotically flat 3-metrics with a prescribed inner boundary. In particular, many examples of asymptotically flat-scalar flat 3-metrics with outermost minimal surfaces are obtained.

1. Introduction

Einstein’s field equation of a space-time is

where is the space-time energy momentum tensor. The equation admits Cauchy data formulation and initial data cannot be chosen arbitrarily. Let be a solution and consider a space-like hypersurface . From the Gauss and Codazzi equations the scalar curvature and the second fundamental form of will satisfy the following constraint equations [15]:

where is the future time-like unit normal vector of the hypersurface When , these equations are called the vacuum constraint equations. There are various ways to construct solutions of the constraint equations. In 1993 Bartnik [3] introduced a new construction of 3-metrics of prescribed scalar curvature and prescribed the inner boundary using a quasi-spherical ansatz. A manifold is called quasi-spherical if it can be foliated by round spheres. Under quasi-spherical ansatz, the equation for the scalar curvature can be written as a semilinear parabolic equation. Let be a smooth compact surface without boundary. Let be equipped with a quasi-spherical metric

for some functions and where is the standard metric on the unit 2-sphere. By viewing as an unknown function and the scalar curvature of and as prescribed fields, Bartnik [3] observed that the function is given by

where denotes the covariant derivative of , , and .

Bartnik’s parabolic method under quasi-spherical ansatz has been generalized by several authors. In 2002, Shi and Tam [10] used the foliation by level sets of the distance function to a convex hypersurface in . Let be a smooth compact strictly convex hypersurface in , and the distance function from . The metric on is of the form

where is the induced metric on , which is the hypersurface with distance from The function with prescribed flat-scalar curvature satisfies the equation

where is the mean curvature of in , is the scalar curvature of , and is the Laplacian on Shi and Tam showed that for a smooth compact strictly convex hypersurface in , a positive function can be arbitrarily prescribed initially.

Weinstein and Smith studied this parabolic method under quasi-convex foliations in [13, 14]. A topological 2-sphere is said to be quasi-convex if its Gauss and mean curvature are positive. A foliation is quasi-convex if its leaves are quasi-convex spheres. Using the Poincaré Uniformization, on can be written as

where are local coordinates on a topological 2-sphere and is a fixed round metric of area . The parabolic equation for on is given by

where , and , , and are functions that can be calculated in terms of only and Weinstein and Smith derived conditions on the source functions , and from above that guarantee the existence of a global positive solution on the interval . However, sometimes the decay conditions may not be verified.

For initial data with an apparent horizon, the parabolicity of the parabolic equation breaks down on the horizon. To overcome this, Smith [12] considered the metrics of the form

on , where is a given Riemannian surface, satisfies

where is the standard metric on and the scalar curvature of is positive. The second condition, , allows the separation of variables so that solving the parabolic scalar curvature equation reduces to solving the elliptic equation

on the region which is referred as the collar region. He then obtained asymptotically flat time symmetric initial data on by constructing the metric on the collar region, and the metric exterior to the collar region using the parabolic method.

This parabolic construction also provides an insight into the extension problem, which is suggested by the definition of quasi-local mass [2]. Here one hopes to extend a bounded Riemannian 3-manifold to an asymptotically flat 3-manifold with nonnegative scalar curvature containing isometrically. The condition that the scalar curvature can be defined distributionally and bounded across leads to the geometric boundary conditions

where the metrics and the mean curvatures in and match along the boundary . Note that this extension is not possible with the traditional conformal method [4, 9]. Specifying both the boundary metric and the mean curvature leads to simultaneous Dirichlet and Neumann boundary conditions, which are ill-posed for the elliptic equation of the conformal factor.

In this paper, we let be a family of metrics on a topological sphere where is the standard metric on the sphere. Let , be a given function on , and

be a metric on with scalar curvature . The metric has the scalar curvature if and only if satisfies the parabolic equation

where and denote the scalar curvature and the Laplacian with respect to , respectively.

First, we study decay conditions of the foliation and prescribed scalar curvature which ensure the solution gives an asymptotically flat metric with prescribed scalar curvature (Theorem 1). Second, with suitable decay conditions we show that there exists a solution so that the metric has outermost totally geodesic boundary. Instead of assuming a collar region, we use the dilation invariance of weighted Hölder norms together with suitable curvature conditions to obtain uniform bounds of solutions to the initial value problem (1) with initial condition on . By Arzela-Ascoli Theorem, there exists a weak solution to (1) with (Theorem 2). Since the mean curvature of stays positive, by the maximum principle the boundary surface is the outermost minimal surface. Theorem 1 and Theorem 2 together give an initial data set of prescribed geometry with a horizon. Last, we study existence results under Ricci flow foliation. It is known by the work of R. Hamilton [6] and B. Chow [5] that the evolution under Ricci flow of an arbitrary initial metric on a topological 2-sphere, suitably normalized, exists for all time and converges exponentially to a round metric. Given a compact Riemannian surface , let

be a metric on where evolves by the Ricci flow defined in (5). Using the fast convergence property, we have corresponding existence results (Theorem 3 and Theorem 4) under Ricci flow foliation. Since are nearly round, the ADM mass of the asymptotically flat manifold is approached by the Hawking mass as (see [11]). If in addition the prescribed scalar curvature is nonnegative, by the equation (11) of , a direct computation shows that

where is the trace-free part of the Ricci potential. We obtain an interesting byproduct, the Hawking mass is nondecreasing in . If we impose the initial condition , i.e., minimal boundary surface, and assume that , then ADM mass is bounded below by

In particular, if we start with the standard metric and prescribe flat scalar curvature , then the metric obtained from above would be exactly a Schwarzschild metric with ADM mass (Corollary 14).

Let for any interval . For compact intervals the parabolic Hölder space is the Banach space of continuous functions on with finite weighted norm, and for noncompact, is defined as the space of continuous functions which are norm-bounded on compact subsets of is the Hölder space on with norm For any , we define functions and by

Our main theorems in this paper are the following:

Theorem 1.

Assume and such that

and

Suppose there exists a constant such that for all and ,

Further assume the nonnegative constant defined by

satisfies

Then for every satisfying

there is a unique positive solution of (1) with the initial condition

(2)

such that the metric on satisfies the asymptotically flat condition

(3)

with ADM mass of that can be expressed as

Moreover the Riemannian curvature of the 3-metric on is Hölder continuous and decays as .

Theorem 2.

Let and be given such that

and

are finite on . Further suppose that for all ,

and

Then there is such that the metric on has curvature uniformly bounded on with totally geodesic boundary.

Let be such that

(4)

Then there is such that

which gives

The modified Ricci flow is defined by the geometric evolution equation

(5)

where is the scalar curvature, is the mean scalar curvature, and the Ricci potential is a solution of the equation with mean value zero. The equation of satisfies

is the trace-free part of . The solution under the modified Ricci flow of an arbitrary initial metric on a topological 2-sphere exists for all time and converges exponentially to the round metric, and exponentially [5, 6]. Moreover if at the start, it remains so for all time. The modified Ricci flow also preserves area. For convenience we normalize the area so that . By the Gauss-Bonnet formula the mean scalar curvature . The solution to the modified Ricci flow provides a canonical foliation on .

Let be a -metric on and be the solution of the Ricci flow defined by (5). Let and be the scalar curvatures of and , respectively. Theorems 3 and Theorem 4 are existence results analogue to Theorems 1 and 2, but under Ricci flow ansatz.

Theorem 3.

Assume that and the constant is defined by

Suppose there is a constant such that for all and ,

Then for any function satisfying

there is a unique positive solution of the parabolic equation

with initial condition such that the metric on satisfies the asymptotically flat condition with finite ADM mass and

Moreover, the Riemannian curvature of the 3-metric on is Hölder continuous and decays as .

Theorem 4.

Let Suppose that for . Let be such that

Then there is a solution such that the constructed metric on has curvature uniformly bounded on with totally geodesic boundary .

The outline of the paper is as follows: In Section 2, we derive the parabolic equation for and its equivalent forms. In Section 3, we prove Theorem 1 in two steps. First, we prove the existence of (Theorem 7). Second, we discuss decay conditions for and metrics which ensure asymptotic flatness of (Theorem 9). After we prove Theorem 1, we show there exists a solution such that the boundary surface is the outermost minimal surface (Theorem 2). In Section 4, we prove similar existence results under Ricci flow foliations.

2. Curvature calculations

For now we use to denote a family of metrics on . Let be a metric on , and and denote the scalar curvatures of and respectively. Let and denote the mean curvature and the norm squared of the second fundamental form of . A direct computation shows that the Ricci curvature of is given by

where is the Laplacian with respect to . The Gauss equation gives that

Combining the above two equations, the scalar curvature with a metric of the form

is given by

(6)

The second fundamental forms measure the change of the metric along the normal direction .

When the second fundamental forms are

In particular,

From (6) and above, we have

where is the Laplacian with respect to . We can rewrite it as

Introducing and , we have the equivalent forms

(7)

and

(8)

where and are the Laplacian and the covariant derivatives on with respect to .

For Ricci flow ansatz, we consider equipped with the metric

where is the solution to the modified Ricci flow (5). Direct computation shows that the second fundamental forms on with respect to the normal are given by

the mean curvature and the norm squared of the second fundamental form are

(9)

respectively. By equation (6), the scalar curvature of is given by

where is the Laplacian with respect to . The metric has the scalar curvature if and only if satisfies the parabolic equation

(10)

The equations for the corresponding terms and are as follows

(11)

and

(12)

where and are the Laplacian and the covariant derivatives on with respect to .

3. Existence

In this section we use the maximum principle to obtain estimates for (Proposition 5), and prove Schauder estimates for and (Proposition 6). Using these a priori estimates, we prove long-time existence of solution Then we show that under suitable decay conditions of the foliation and the scalar curvature , the metric is asymptotically flat and has finite ADM mass. We basically follow the argument in [3], see also [10].

Proposition 5.

Suppose , is a positive solution to (1). Then for we have

and

If we further assume that is defined on such that the functions

and

are defined and finite for all , then the estimates may be rewritten as

and

Proof.

Applying the parabolic maximum principle to (7) for gives

Solving the associated O.D.E., we have