The Cauchy Problem for Membranes

The Cauchy Problem for Membranes

Abstract

We show existence and uniqueness for timelike minimal submanifolds (world volume of -branes) in ambient Lorentz manifolds admitting a time function in a neighborhood of the initial submanifold. The initial value formulation introduced and the conditions imposed on the initial data are given in purely geometric terms. Only an initial direction must be prescribed in order to provide uniqueness for the geometric problem. The result covers non-compact initial submanifolds of any codimension. By considering the angle of the initial direction and vector fields normal to the initial submanifold with the unit normal to the foliation given by the time function we obtain a quantitative description of “distance” to the light cone. This description makes it possible to treat initial data which are arbitrarily close to the light cone. Imposing uniform assumptions give a lower bound for a notion of “time of existence” depending only on geometric quantities involving the length of timelike curves lying in the solution.

ams:
35L15, 35L70, 81T30
1

1 Introduction

Membranes are spacelike submanifolds evolving in a Lorentzian manifold. The equation of motion of a membrane is determined by the condition that the world volume of the membrane, the timelike submanifold swept out during time evolution, is a critical point for the volume functional induced by the ambient manifold. Membranes arise in the context of higher-dimensional extensions of String Theory, where they are called -branes according to the dimension of the spacelike object. The Euler-Lagrange equation of the volume functional is the vanishing of the mean extrinsic curvature vector of the world volume. This equation will be called the membrane equation in the sequel. Due to the signature of the world volume the membrane equation leads to a system of nonlinear wave equations. Since the membrane equation is invariant under diffeomorphisms of the world volume, it is degenerate.

In this paper we consider the solvability of the initial value problem (IVP) for the membrane equation with initial data only involving geometric quantities. Our focus is on providing short-time existence and uniqueness for the Cauchy problem admitting non-compact initial data, and data which are arbitrarily close to the light cone. The Cauchy problem for closed strings, i. e. a membrane diffeomorphic to the circle line, has been studied in Minkowski space and in globally hyperbolic ambient manifolds by T. Deck [1], and O. Müller [2]. Solutions are obtained as timelike immersions in a neighborhood of the initial data. In these works the question of global-in-time existence was investigated for globally hyperbolic ambient manifolds. Exact solutions of this equation corresponding to pulsating and rotating objects have been studied by H. Nicolai and J. Hoppe in [3].

In the case of the world volume in Minkowski space represented as a graph, global existence for small initial data was shown in [4]. This result was generalized to arbitrary codimensions in [5].

Using techniques previously applied to the Einstein equation by D. Christodoulou and S. Klainerman (cf. [6]), S. Brendle investigated in [7] the stability of a hyperplane satisfying the membrane equation in Minkowski space. In contrast to the work of H. Lindblad (cf. [4]), S. Brendle requires less regularity of the initial data, comparable to that in [6].

Let us state the main problem addressed by this paper in non-technical terms.
Let be an -dimensional Lorentzian manifold, and let be an -dimensional spacelike submanifold of . Assume to be an unit timelike future-directed vector field normal to .

Existence

Find an open -dimensional timelike submanifold of satisfying

(1)

Uniqueness

Show that for and solving the IVP (1), there exists an open set with such that

We will refer to , and as ambient manifold, initial submanifold, and initial direction, respectively. The main result of this work is an affirmative answer to the problem above under suitable conditions. Here, the ambient manifold is assumed to admit a time function in a neighborhood of the initial submanifold. No further condition on the causal structure is required.

After fixing some notation in section 2, we perform the reduction of the membrane equation to a system of quasilinear hyperbolic equations by employing a gauge used in the context of the Einstein equations (cf. [8]). In section 4 we consider solutions of the membrane equation given by parametrized immersions. In section 5 these local solutions are glued together to obtain a solution to the geometric Cauchy problem (1). The main existence result, Theorem 5.2, shows that for non-compact regularly immersed submanifolds of any codimension in an ambient manifold of any dimension, existence and uniqueness holds. Uniform assumptions lead to a lower bound on the length of timelike curves lying in the world volume, which gives rise to a geometric notion of time of existence of a solution to the Cauchy problem (1). In Corollary 5.6 it is shown that smooth data lead to smooth solutions of the membrane equation. In Theorem 5.7 we present the main uniqueness result for the membrane equation in the case of smooth data.

The results are taken from the author’s Ph.D. thesis; for further reference see [9].

2 Notation

Let be an -dimensional smooth manifold endowed with a Lorentzian metric . The Levi-Civita connection with respect to and the corresponding Christoffel symbols are denoted by and , respectively. Covariant derivatives of order are denoted by . The or the -version of the curvature of is denoted by .

Suppose is an -dimensional timelike submanifold of . The metric and connection induced on are then given by

The Christoffel symbols of are denoted by . The second fundamental form and the mean curvature of are given by

(2)

The usual definition of the mean curvature involves a factor dependent on the dimension, which was omitted here since we are only interested in the homogeneous equation.

The following definitions introduce types of initial submanifolds. We refer to them as the immersion type of the initial submanifold.

Definition 2.1

Let be an -dimensional manifold, and let be an immersion. The image of , , is called a regularly immersed submanifold of .

Definition 2.2

Let be a regularly immersed submanifold of dimension with immersion . The set is called a locally embedded submanifold if for every point there exist open sets and such that

(3)
Definition 2.3

Let be a regularly immersed submanifold of dimension with immersion . The set is called a regularly immersed submanifold with finite intersections if for every point there exist a neighborhood of and finitely many open disjoint sets such that

(4)

Let be a regularly immersed submanifold with immersion . We make use of the induced metric on . The Levi-Civita-connection compatible with is denoted by . The connection on the pull-back bundle is denoted by , and members of are called vector fields along . Using this notation we introduce the second fundamental form of defined as the vector field

(5)

where denotes the differential of , and denotes the part normal to the image of . If is a locally embedded submanifold then this definition is independent of the local embedding and gives the usual definition of second fundamental form of a submanifold.

Let be a Riemannian metric defined on . For a vector field along we introduce the following norm

(6)

with the abbreviation . Here, we used the following set of indices for coordinates. On and on the initial submanifold , coordinates carry small Latin indices running from to . Capital Latin indices as will run from to , and indicate coordinates on . Our convention for the signature of a Lorentzian metric is . Partial derivatives in coordinates are abbreviated by . Local coordinates on the timelike submanifold carry Greek indices running from to .

An assumption on the causal structure of the ambient manifold will be made by using the following terminology.

Definition 2.4

A real-valued function on is called time function if its gradient is everywhere timelike. Such a time function induces a time foliation of by its levelsets , which are spacelike hypersurfaces.

We further introduce the lapse of the foliation given by by

(7)

and the future-directed unit normal to the time foliation is given by

The dual vector field to the differential of the time function is given by

The existence of such a function provides us with a possibility of introducing a Riemannian metric.

Definition 2.5

Suppose admits a time function . We consider the decomposition into tangential and normal part with respect to the slices, denoted by and , respectively. We introduce a Riemannian metric associated to the time foliation by a change of the sign of the unit normal as follows

(8)

Since the slices are spacelike this metric is Riemannian.

We use the norm on tensors induced by the metric defined as follows

3 Reduction

In this section we present a method to obtain a strictly hyperbolic system from the membrane equation by using parametrizations and a specific gauge condition.

Let denote an immersion with induced metric . We denote the Christoffel symbols with respect to by . The image of will correspond to a solution of the geometric IVP (1).

Let be coordinates on , and let be coordinates on . We denote the covariant derivative on the pullback bundle by . The membrane equation for the image of then reads in this situation

(9)
or, equivalently (10)

where denotes the part tangential to the image of .

The hyperbolicity of (10) comes from the signature of the metric induced by on . The degeneracy of (10) is a consequence of the invariance under tangential diffeomorphisms of a solution. It becomes manifest in the contracted Chrisoffel symbols induced on , which contain a term involving second-order derivatives of cancelling the tangential part of the leading second-order term.

To overcome the problem of degeneracy it is necessary to fix a gauge. We consider the so-called harmonic map gauge taken from [8], where it is used to show well-posedness of the Einstein equations.

Recall that a map between two pseudo-Riemannian manifolds is called harmonic map (or wave map) if it satisfies

(11)
Definition 3.1

A solution of the membrane equation (10) is in harmonic map gauge, if the following condition is satisfied

(12)

where is a fixed background metric. We call this condition harmonic map gauge condition.

Remark 3.2

In coordinates, condition (12) reads

(13)

Note that this is the trace of the difference of two connections, and is therefore independent of coordinates.

By inserting the harmonic map gauge condition (13) into (10) we obtain the reduced membrane equation

(14)

The reduced equation is strictly hyperbolic since the contracted Christoffel symbols are replaced by a lower-order term. We now show equivalence of the equations (14) and (10).

Lemma 3.3

The membrane equation (10) together with condition (13) is equivalent to (14).

Proof:
We only need to show that the reduced membrane equation (14) implies (10) and condition (13); the other direction follows immediately. Suppose the reduced membrane equation (14) holds. We compute the contracted Christoffel symbols with respect to the metric induced by a solution

Using this identity together with (14) gives us the desired result. □

Remark 3.4

This result can be derived in a somewhat more abstract fashion if we use (9) instead of (10). The reduced membrane equation (14) can be expressed

The right hand side of the latter equation is tangential to the image of , therefore the normal part of the left hand side has to vanish, and the tangential part has to coincide with the right hand side which yields the desired identity (13).

Remark 3.5

Other gauges corresponding to the prescription of the contracted Christoffel symbols are possible. Therefore, we can mine the rich repertoire of gauges for the Einstein equations (cf. [8]).

Remark 3.6

In contrast to the case of the Einstein equations the reduction process needs no further equation.

Based on (10) we consider the following problem involving existence and uniqueness.
Let be a regularly immersed -dimensional submanifold of with immersion . Let be a function on called initial lapse, and let be a vector field on called initial shift. Suppose is an unit timelike future-directed vector field along normal to .

Existence

Find a neighborhood of and an immersion solving the IVP

(15)

such that is timelike, and the image of is spacelike. The last term consists of the differential of applied to the vector field . The parameter denotes the first component of .

Uniqueness

Suppose is locally embedded, and let be an unit timelike future-directed vector field on . Assume to be an immersion with , and let and be another choice of initial lapse and shift.

Show that for immersions solving (10), and attaining the initial values

there exists a local diffeomorphism of such that .

Remark 3.7

Throughout the following we use a special background metric defined on . If the initial values of are and , then we define

(16)

Observe that , and are independent of the parameter .

4 Parametrized Immersions

This section is devoted to discuss the Cauchy problem (15). Solutions will be obtained as parametrization through spacelike immersions. First we present the main assumptions of this paper. The main results of this section are stated in Propositions 4.2 and 4.6.

4.1 Assumptions

We use the assumptions on the date presented in this section if not explicitly stated otherwise. In order to get results which are independent of the scale of the ambient manifold — multiplication of the metric with the square of a positive constant — we introduce a constant . The purpose of this constant is to absorb scaling such that scale-invariant constants occur in the assumptions.

Let be an integer, and let and denote open sets.
Assumptions on the ambient manifold: We suppose to admit a time function in . Assume the metric and to be . Let denote the lapse of the time foliation induced by defined by (7), and let denote the Riemannian metric introduced in Definition 2.5.

We suppose there are constants , and independent of such that the following inequalities are uniformly satisfied in

where denotes the -tensor obtained by applying the covariant derivative to the gradient of the time function .
Assumptions on the initial submanifold: Assume to be . Let admit a time function in a neighborhood of . We suppose there exist constants independent of such that the following inequalities are uniformly satisfied in

where denotes the unit timelike future-directed normal to the time foliation on , and denotes the second fundamental form of (cf. (5)).
Assumptions on the initial direction: Assume to be . We suppose there exist constants independent of such that the following inequalities are uniformly satisfied in

Remark 4.1

If the initial submanifold is locally embedded, and the initial direction is defined on rather than on the domain of the embedding, then the assumptions on and do not depend on the local embedding.

When considering the initial value problem (15) for immersions we have the addinitional assumptions on initial lapse and shift.
Assumptions on initial lapse and shift: Assume and to be . We suppose there exist constants , and independent of such that the following inequalities are uniformly satisfied in

4.2 Existence

The following result gives a solution to the IVP (15). For a detailed proof we refer to [9].

Proposition 4.2

Suppose for each there exist a neighborhood of , and an open set containing such that the assumptions described in the previous section are satisfied in and .

Then there exist a neighborhood of and a -immersion such that is timelike, and the image of is spacelike solving the reduced membrane equation (14) with respect to the background metric defined by (16), and attaining the initial values and .

Remark 4.3

The proof involves solving the reduced membrane equation (14) in a neighborhood of each point . Standard local uniqueness results for hyperbolic equations then show that two local solutions coincide on their common domain. Therefore, a spatially local version of Proposition 4.2 is also valid.

Remark 4.4

Let be an integer, and suppose . Then the solution is of class .

Remark 4.5

If the assumptions are uniform in , then the domain of the solution has the form with a constant independent of the scale.

4.3 Uniqueness

Let be a locally embedded submanifold of . Suppose are two immersions satisfying the conditions of Definition 2.2 for . The metrics induced on by we denote by , respectively. Let denote the local diffeomorphism defined by . The differential of is a member of , and we use the following norm

Norms of higher-order derivatives of are defined analogously. Let be two functions on , and let be two vector fields on . Assume to be defined on .

We obtain the following uniqueness result.

Proposition 4.6

Let and satisfy the assumptions of sections 4.1. Suppose there exist constants independent of such that

(17)

Let be two -solutions of the membrane equation (10) attaining the initial values

Let be the background metric defined by (16) using the initial values of . Assume to be in harmonic map gauge with respect to .

Then there exists a local diffeomorphism of such that and coincide on a neighborhood of .

From [9] we have the following uniqueness result for solutions to the IVP (15), which are in harmonic map gauge with respect to the background metric defined by (16).

Proposition 4.7

Let the assumptions of section 4.1 be satisfied. Assume and to be two immersions of class solving the IVP (15). Suppose both solutions are in harmonic map gauge with respect to the background metric defined by (16).

Then there exists a neighborhood of , on which the two solutions and coincide.

Remark 4.8

If the domains and have the form , then there exists a scale-invariant constant such that and coincide for .

Remark 4.9

As is the case for the existence result stated in Proposition 4.2, a spatially local version of Proposition 4.7 holds as well. From uniqueness results for hyperbolic equations, we obtain that uniqueness holds on cones.

The previous result relies on the fact that both solutions are in harmonic map gauge with respect to the same background metric. The following proposition gives a condition under which it is possible to align the harmonic map gauge of two solutions.

Proposition 4.10

Suppose in the situation of Proposition 4.6 there exist open sets such that for a fixed point it holds and . Suppose there exists a diffeomorphism such that

(18)

Furthermore, assume the inverse to satisfy the initial conditions

(19)
with

Then and coincide on a neighborhood of .

Proof:
From the harmonic map equation satisfied by we derive the condition for the solution of the membrane equation to be in harmonic map gauge with respect to the background metric (cf. (12)).

It remains to show that the initial values of give us the appropriate initial values for . From the definition of we derive that . To obtain the initial value for the velocity we compute

Since the second and the last term cancel, the claim follows from the definition of the initial values of . Therefore, and satisfy the reduced membrane equation (14) with respect to the background metric attaining the initial values of . From Proposition 4.7 we obtain that both immersions coincide locally. □

To obtain Proposition 4.6 it remains to show existence of a local diffeomorphism satisfying condition (18). Since the metrics and are Lorentzian, the harmonic map equation for has a structure similar to the reduced membrane equation (14). The main difference is that the harmonic map equation is semilinear. Hence, the same techniques can be applied as to obtain a solution, and we have the following result.

Proposition 4.11

In the situation of Proposition 4.6 there exist a neighborhood of and a -immersion satisfying (11) with and . Furthermore, is invertible in a neighborhood of , and the inverse attains the initial values (19).

5 Geometric results

In this section we consider existence and uniqueness for the membrane equation in purely geometric terms. An existence result is obtained including a geometric notion of time of existence of a solution. The uniqueness result shows that two submanifolds solving the IVP coincide on a neighborhood of the initial submanifold.

5.1 Existence

The next definition gives a notion of “time of existence” which respects the geometric behavior of a solution; note this does not necessarily coincide with the time parameter of Proposition 4.2.

Definition 5.1 (Time of existence)

Let be a solution of the IVP (1). The time of existence of is given by

future-directed

Our aim is an existence result for the IVP (1) which includes a lower bound on the time of existence. Uniform assumptions imposed on ambient manifold, initial submanifold, and initial direction provide us with such a lower bound.

Theorem 5.2

Let be a constant. Suppose for each there exists a neighborhood of such that , and the assumptions of section 4.1 are satisfied in and with constants independent of .

Then there exists an open -dimensional regularly immersed Lorentzian submanifold of class satisfying the IVP (1). Furthermore, there exists a scale-invariant constant such that

Proof:
From Proposition 4.2 and Remark 4.5 we obtain an immersion of class solving the IVP (15) with initial lapse equal 1 and initial shift equal 0. Letting gives us a regularly immersed timelike submanifold solving the IVP (1).

The construction of provides us with a lower bound on the time parameter , and an estimate for the timelikeness of . Hence, a lower bound exists for the length of the timelike curves for all . □

Remark 5.3

The proof shows that the theorem applies to the situation where the assumptions are valid only locally.

The following corollaries show that a solution to the IVP (1) for the membrane equation can be constructed in such a way that the immersion type of the initial submanifold is preserved.

Corollary 5.4

Let be locally embedded; for each point let and denote sets satisfying the conditions (3) of Definition 2.2. Suppose that for each point the assumptions of section 4.1 are satisfied in and .

Then there exists a locally embedded timelike -dimensional submanifold of class solving the IVP (1).

Proof:
A local solution to the IVP (15) with initial lapse equal 1 and initial shift equal 0 can be obtained by a local version of Proposition 4.2 (cf. Remark 4.3). This solution is defined on a neighborhood of with values in . From the inverse function theorem we obtain that the solution is an embedding in a neighborhood of . By shrinking the domain of further we achieve that the set lies in . Consider the family of the domains of all local solutions. The intersection with provides a covering of . Choose points such that the intersection of the family of domains of the local solutions with form a locally finite covering subordinate to the above covering. Define a mapping which coincides with the local solutions on each set belonging to the locally finite subcovering above. From a local version of Proposition 4.9 we derive that local solutions coincide on their common domain. Therefore, this mapping is a well-defined immersion, whose image is a locally embedded timelike submanifold of class . □
To obtain a solution for regularly immersed initial submanifolds with locally finite intersections it is necessary to change the way in which the local solutions are pieced together.

Corollary 5.5

Let be regularly immersed with locally finite intersections; for each point let and denote sets satisfying the conditions (4) of Definition 2.3. Suppose that for each point the assumptions of section 4.1 are satisfied in and for every .

Then there exists a timelike -dimensional regularly immersed submanifold with locally finite intersections of class solving the IVP (1).

Proof:
For each we pick the finitely many satisfying (4), and solve the membrane equation in with values in . We shrink the domain of the solutions to obtain embeddings. By shrinking the set to a subset we achieve that

Consider the family . Choose a locally finite covering of subordinate to . Let denote the part of which belongs to . Consider the family . Then the family is a locally finite covering of subordinate to due to the finiteness of the sets for fixed .

Let be a mapping defined by on . Since the local solutions coincide on common domains, this mapping is well-defined. By construction it follows that is regularly immersed with locally finite intersections and is of class . □

We show that smooth data lead to a smooth solution of the IVP (1) for the membrane equation again respecting the immersion type of the initial submanifold.

Corollary 5.6

Assume to be smooth, and suppose is either smoothly

  1. regularly immersed,

  2. locally embedded, or

  3. regularly immersed with locally finite intersections.

Suppose admits a smooth time function in a neighborhood of the initial submanifold . Assume to be smooth.

Then there exists an open smooth -dimensional timelike submanifold which is

  1. regularly immersed,

  2. locally embedded, or

  3. regularly immersed with locally finite intersections,

satisfying the IVP (1).

Proof:
For each integer it follows from the smoothness of , the immersion , and the initial direction that the assumptions of section 4.1 are satisfied for in a neighborhood of each point . Depending on the immersion type of we apply Theorem 5.2 and Remark 5.3, or Corollaries 5.4 and 5.5. We obtain a solution of class by taking Remark 4.4 into account. □

5.2 Uniqueness

In this section we consider the geometric uniqueness problem. In Proposition 4.6 it was shown that the construction of a solution made in Proposition 4.2 is independent of the choice of immersion of the initial submanifold, initial lapse, and initial shift. Therefore, it remains to construct an immersion of an arbitrary solution to (1) which is in harmonic map gauge with respect to the background metric defined by the initial values.

Theorem 5.7

Assume to be smooth, and suppose is smooth locally embedded. Let admit a smooth time function in a neighborhood of the initial submanifold . Suppose the initial direction is smooth.

Let and be two open smooth -dimensional locally embedded Lorentzian submanifolds of solving the IVP (1).

Then there exists a neighborhood of such that

Remark 5.8

In view of the fact that the image of a solution to (15) is a solution to (1) (cf. proof of Theorem 5.2), Theorem 5.7 and Proposition 4.6 show that initial lapse and shift can be given freely.

Our strategy to prove this theorem is to compare an arbitrary solution with the solution constructed in the previous section. To apply the uniqueness result of Proposition 4.6 we need to construct an immersion satisfying the IVP (15).

Proposition 5.9

Let , and satisfy the assumptions of Theorem 5.7. Let be a smooth locally embedded solution to the IVP (1).

Then there exists an immersion with a locally embedded submanifold. Furthermore, has the properties that is timelike, and has a spacelike image. The initial values of are given by and .

Proof:
Let , and let be a geodesic in attaining the initial values and . Set . Then is an immersion, since is assumed to be an immersion, and is assumed to be unit timelike. The initial values of follow from the initial values of the geodesics. From the smoothness of the initial data it follows that is also smooth.

The construction above is similar to Gaussian coordinates. In an analogous way it follows that is timelike, and has a spacelike image. From the same argument we derive that the geodesics do not cross in a neighborhood of any point in as long as is an embedding, and is an embedded submanifold around that point. □

We are now in the position to give a proof of the main uniqueness result.
Proof of Theorem 5.7:
Let be the solution to the IVP (15) with initial values and constructed in Corollary 5.6. Let denote the locally embedded image of the solution . We compare a smooth solution to the IVP (1) with the solution .

From Proposition 5.9 we obtain that admits an immersion with locally embedded image, and initial values and in a neighborhood of the initial submanifold .

Proposition 4.6 now yields that there is a local diffeomorphism such that and coincide. Hence, the desired result follows. □

The author would like to thank Gerhard Huisken for interesting discussions and encouragement, and Paul Allen for helpful assistance.

References

Footnotes

  1. : Class. Quantum Grav.

References

  1. Thomas Deck. A geometric Cauchy problem for timelike minimal surfaces. Ann. Global Anal. Geom., 12(4):305–312, 1994.
  2. Olaf Müller. The Cauchy problem of Lorentzian minimal surfaces in globally hyperbolic manifolds. Ann. Global Anal. Geom., 32(1), 2007.
  3. Jens Hoppe and Hermann Nicolai. Relativistic minimal surfaces. Phys. Lett. B, 196(4):451–455, 1987.
  4. Hans Lindblad. A remark on global existence for small initial data of the minimal surface equation in Minkowskian space time. Proc. Amer. Math. Soc., 132(4):1095–1102 (electronic), 2004.
  5. Paul Allen, Lars Andersson, and James Isenberg. Timelike minimal submanifolds of general co-dimension in Minkowski space time. J. Hyperbolic Differ. Equ., 3(4):691–700, 2006.
  6. D. Christodoulou and S. Klainerman. The Global Nonlinear Stability of the Minkowski Space, volume 41 of Princeton Mathematical Series. Princeton University Press, Princeton, New Jersey, 1993.
  7. Simon Brendle. Hypersurfaces in Minkowski space with vanishing mean curvature. Comm. Pure Appl. Math., 55(10):1249–1279, 2002.
  8. Helmut Friedrich and Alan Rendall. The Cauchy problem for the Einstein equations. In Einstein’s field equations and their physical implications, volume 540 of Lecture Notes in Phys., pages 127–223. Springer, Berlin, 2000.
  9. Olaf Milbredt. The Cauchy problem for membranes. PhD thesis, Free University Berlin, Berlin, 2008. arXiv:0807.2539v1 [gr-qc].
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