Minimal surfaces in flat and curved spacetimes of arbitrary dimensionality

Minimal surfaces and domain walls play important roles in astrophysics, general relativity, and modern particle theories. First, we review the Bernstein theorem, which states that a well-defined single-valued minimal surface in 3-dimensional Euclidean flat spaces should be a plane, and its failure in general dimensions. Then, we show that this failure of Bernstein theorem in higher dimensions results in interesting consequences in the systems where the branes interact with black holes. 1 Bernstein conjecture, branes, and minimal cones Minimal surfaces in Euclidean space E have been extensively studied since the pioneering work of Thomas Young and of Laplace. In Monge, or non-parametric, gauge the surface is specified by the height function z = z(x, y) above some plane. The non-parametric minimal surface equation governing the function z(x, y) is ∂x  zx √ 1 + z2 x + z2 y  + ∂y  zy √ 1 + z2 x + z2 y  = 0 . (1) A famous result of Bernstein asserts that the only single valued solution of Eq. (1) defined for all (x, y) ∈ R is a plane. It may also be shown that the planar solution is a minimizer of the area functional among compactly supported variations of the surface. In terms of brane theory, this means that the “classical ground state”, i.e., the static minimum of the energy functional for a membrane in three dimensional Euclidean space E, which may be thought of as a static configuration in 4-dimensional Minkowski spacetime E, is smooth and indeed planar. From the world volume point of view the classical ground state of the membrane preserves (2+1)-dimensional Poincaré invariance and may be thought of as a copy of (2+1)-dimensional Minkowski spacetime E. It is natural to conjecture that Bernstein’s theorem remains valid for a minimal p-dimensional hypersurface in (p + 1)-dimensional Euclidean space E. In other words, the classical ground state of a p-brane in (p + 1 + 1)-dimensional Minkowski spacetime E should be flat and invariant under the action of the (p + 1)-dimensional Poincaré group E(p, 1). Remarkably, although true for p ≤ 7 it fails for p+1 = 9 [2]. In other words, the classical ground state of an 8-brane in 10-dimensional Minkowski spacetime spontaneously breaks 10-dimensional Poincaré invariance. The proof [2] rests on the fact that in E and above, a minimal hypersurface which is a minimizer of the p-volume functional among compactly supported variations need not be smooth. There are rather explicit counterexamples called minimal cones. Their existence leads to the conclusion that Bernstein’s theorem fails in E [2]. As far as we aware, there has been very little discussion of the significance of this fact in the M/String theory literature. The breakdown of regularity of minimal hypersurfaces of flat space extends to minimal hypersurfaces of curved Riemannian manifolds and this has consequences for proofs of the positive energy theorem of general relativity which make essential use of minimal surfaces as a technical tool (e.g., see [3] for the application to the positive mass theorem). It seems worthwhile therefore to examine the behavior of minimal surfaces in higher dimensions and in curved spaces in some explicit detail in order to understand better the situation and its possible physical implications. In particular, it is interesting to see whether the existence of various critical dimensions which has been noted in related contexts is of 1Email address: umpei@rikkyo.ac. jp 2This article is based on paper [1]. 2 Minimal surfaces in flat and curved spacetimes of arbitrary dimensionality a universal nature and related to the the breakdown of Bernstein’s theorem and the existence of minimal cones. To make progress it is helpful to assume that the relevant surfaces have sufficient symmetries with which the problem may be reduced to one involving ordinary differential equations in an appropriate quotient space X, a ploy known to mathematicians as equivariant variational theory. Typically the brane equations of motion reduce to finding geodesics in X with respect to a suitable metric g on X, induced by the p-volume functional. The p-brane will be p-volume minimizing if the corresponding geodesic γ is length minimizing. A necessary condition that a geodesic joining points a and b be length minimizing is that γ contains no points between a and b conjugate to either. The existence of such conjugate points is governed by the Jacobi or geodesic deviation equation, solutions of which depend on the curvature of X. In the case that X is 2-dimensional, it is the sign of the Gauss curvature K which is important. If for example K is negative in the vicinity of γ, then it can contain no conjugate points and hence must be locally length minimizing. In the cases we shall consider the Gauss curvature is actually positive and a more detailed examination is required. One might have thought that positive Gauss curvature would lead to a second variation or Hessian of indefinite sign. However, the situation is more subtle since the effective metric governing the variational principle is incomplete and becomes singular near a conical point and compensatory terms can arise which in low dimension render the Hessian positive definite. 2 Branes in flat spaces The simplest example is when a Lie group G acts isometrically on a (p + 1)-dimensional Riemannian manifold {Σ, h}, thought of as “space” with orbits which are (p − 1)-dimensional. The X = Σ/G is (locally) 2-dimensional and a curve in X may be thought of as the projection under the action of G of a p-dimensional submanifold S of Σ. As a simple but primary example, let us consider a G-invariant submanifold S ⊂ Σ = E (m,n ≥ 1) as the configuration of p-brane (p = n + m + 1). We focus on the case of G = SO(m + 1)× SO(n + 1) ⊂ SO(m + n + 2). The metric of ambient flat space is written in the form h = dx + xdΩm + dy 2 + ydΩn . (2) A projected metric h̃ on Σ/G is defined via a transformation of an inner product, h̃(u1, u2) = h(u1, u2), where (u1, u2) is a pair of vectors tangent to the orbit G(s), s ∈ S. Thus, in the present case h̃ = dx+dy . The volume function on Σ/G, which is the volume of the orbit G(s), is given by v(x, y) := Vol (G(s)) = ΩmxΩny , where Ωn is the volume of unit n-sphere. Now, we are ready to define an effective space where the geodesic γ is to be found. The metric g of the effective space is defined as g = d` := v(x, y)h̃ = ΩmΩ 2 nx y(dx + dy) , (3) where λ is the co-dimension of the G-invariant surface S. For the present example, λ is given by λ := dimS − dimG(x) = (m + n + 1) − (m + n) = 1 . The problem to find the minimal surface has been reduced to find the geodesic γ with g = d`. The action to be minimized is ` = ∫ xy √ dx2 + dy2 . If one denotes the geodesic by y = y(x) and varies the action with respect to it, one has xyy′′ + (myy′ − nx)(1 + y′2) = 0 . (4) One can easily see that a cone y = √ n/m x solves Eq. (4). As noted before, one cannot know whether the above cone is indeed a minimizer or not until examining the second variation. Here, we introduce an alternative criterion to examine whether a geodesic is a minimizer or not. Readers are directed to [4] for an introduction to this topic. The Jacobi equation or equation of geodesic deviation is written as dη d`2 + Kη = 0 , (5) where η and K are the geodesic deviation and Gauss curvature of metric g. For a general metric of form g = v(x, y)(dx + dy), the Gauss curvature is given by K = (1/2)(Ricci scalar) = −2λ−1v−2/λ(∂2 x +