Small Geodesic Balls Research Articles

Overdetermined elliptic problems in nontrivial contractible domains of the sphere

In this paper, we prove the existence of nontrivial contractible domains Ω⊂Sd, d≥2, such that the overdetermined elliptic problem{−εΔgu+u−up=0in Ω, u>0in Ω, u=0on ∂Ω, ∂νu=constanton ∂Ω, admits a positive solution. Here Δg is the Laplace-Beltrami operator in the unit sphere Sd with respect to the canonical round metric g, ε>0 is a small real parameter and 1<p<d+2d−2 (p>1 if d=2). These domains are perturbations of Sd∖D, where D is a small geodesic ball. This shows in particular that Serrin's theorem for overdetermined problems in the Euclidean space cannot be generalized to the sphere even for contractible domains.

Volume comparison with respect to scalar curvature

In this article, we investigate the volume comparison with respect to scalar curvature. In particular, we show volume comparison holds for small geodesic balls of metrics near a V-static metric. For closed manifold, we prove the volume comparison for metrics near a strictly stable Einstein metric. As applications, we give a partial answer to a conjecture of Bray and recover a result of Besson, Courtois and Gallot, which partially confirms a conjecture of Schoen about closed hyperbolic manifold. Applying analogous techniques, we obtain a different proof of a local rigidity result due to Dai, Wang and Wei, which shows it admits no metric with positive scalar curvature near strictly stable Ricci-flat metrics.

Large Steklov eigenvalues via homogenisation on manifolds

Using methods in the spirit of deterministic homogenisation theory we obtain convergence of the Steklov eigenvalues of a sequence of domains in a Riemannian manifold to weighted Laplace eigenvalues of that manifold. The domains are obtained by removing small geodesic balls that are asymptotically densely uniformly distributed as their radius tends to zero. We use this relationship to construct manifolds that have large Steklov eigenvalues. In dimension two, and with constant weight equal to 1, we prove that Kokarev’s upper bound of 8pi for the first nonzero normalised Steklov eigenvalue on orientable surfaces of genus 0 is saturated. For other topological types and eigenvalue indices, we also obtain lower bounds on the best upper bound for the eigenvalue in terms of Laplace maximisers. For the first two eigenvalues, these lower bounds become equalities. A surprising consequence is the existence of free boundary minimal surfaces immersed in the unit ball by first Steklov eigenfunctions and with area strictly larger than 2pi . This was previously thought to be impossible. We provide numerical evidence that some of the already known examples of free boundary minimal surfaces have these properties and also exhibit simulations of new free boundary minimal surfaces of genus zero in the unit ball with even larger area. We prove that the first nonzero Steklov eigenvalue of all these examples is equal to 1, as a consequence of their symmetries and topology, so that they are consistent with a general conjecture by Fraser and Li. In dimension three and larger, we prove that the isoperimetric inequality of Colbois–El Soufi–Girouard is sharp and implies an upper bound for weighted Laplace eigenvalues. We also show that in any manifold with a fixed metric, one can construct by varying the weight a domain with connected boundary whose first nonzero normalised Steklov eigenvalue is arbitrarily large.

Scalar curvature and the relative capacity of geodesic balls

In a Riemannian manifold, it is well known that the scalar curvature at a point can be recovered from the volumes (areas) of small geodesic balls (spheres). We show the scalar curvature is likewise determined by the relative capacities of concentric small geodesic balls. This result has motivation from general relativity (as a complement to a previous study by the author of the capacity of large balls in an asymptotically flat manifold) and from weak definitions of nonnegative scalar curvature. It also motivates a conjecture (inspired by the famous volume conjecture of Gray and Vanhecke), regarding whether Euclidean-like behavior of the relative capacity on the small scale is sufficient to characterize a space as flat.

Pucci eigenvalues on geodesic balls

We study the eigenvalue problem for the Riemannian Pucci operator on geodesic balls. We establish upper and lower bounds for the principal Pucci eigenvalues depending on the curvature, extending Cheng’s eigenvalue comparison theorem for the Laplace–Beltrami operator. For manifolds with bounded sectional curvature, we prove Cheng’s bounds hold for Pucci eigenvalues on geodesic balls of radius less than the injectivity radius. For manifolds with Ricci curvature bounded below, we prove Cheng’s upper bound holds for Pucci eigenvalues on certain small geodesic balls. We also prove that the principal Pucci eigenvalues of an \({O(n)}\)-invariant hypersurface immersed in \({{\mathbb{R}}^{n+1}}\) with one smooth boundary component are smaller than the eigenvalues of an \({n}\)-dimensional Euclidean ball with the same boundary.

Entanglement Equilibrium and the Einstein Equation.

A link between the semiclassical Einstein equation and a maximal vacuum entanglement hypothesis is established. The hypothesis asserts that entanglement entropy in small geodesic balls is maximized at fixed volume in a locally maximally symmetric vacuum state of geometry and quantum fields. A qualitative argument suggests that the Einstein equation implies the validity of the hypothesis. A more precise argument shows that, for first-order variations of the local vacuum state of conformal quantum fields, the vacuum entanglement is stationary if and only if the Einstein equation holds. For nonconformal fields, the same conclusion follows modulo a conjecture about the variation of entanglement entropy.

Localized Lp-estimates of eigenfunctions: A note on an article of Hezari and Rivière

We use a straightforward variation on a recent argument of Hezari and Rivière [8] to obtain localized Lp-estimates for all exponents larger than or equal to the critical exponent pc=2(n+1)n−1. We are able to do this directly by just using the Lp-bounds for spectral projection operators from our much earlier work [12]. The localized bounds we obtain here imply, for instance, that, for a density one sequence of eigenvalues on a manifold whose geodesic flow is ergodic, all of the Lp, 2<p≤∞, bounds of the corresponding eigenfunctions are relatively small compared to the general ones in [12], which are saturated on round spheres. The connection with quantum ergodicity was established for exponents 2<p<pc in the recent results of the author [13] and Blair and the author [3]; however, the article of Hezari and Rivière [8] was the first one to make this connection (in the case of negatively curved manifolds) for the critical exponent, pc. As is well known, and we indicate here, bounds for the critical exponent, pc, imply ones for all the other exponents 2<p≤∞. The localized estimates involve L2-norms over small geodesic balls Br of radius r, and we shall go over what happens for these in certain model cases on the sphere and on manifolds of nonpositive curvature. We shall also state a problem of determining when one can improve on the trivial O(r12) estimates for these L2(Br) bounds. If r=λ−1, one can improve on the trivial estimates if one has improved Lpc(M) bounds just by using Hölder's inequality; however, obtaining improved bounds for r≫λ−1 seems to be subtle.

A characterization of spaces of constant curvature by minimum covering radius of triangles

In this paper, we prove that if in a Riemannian manifold, the minimum covering radius of a point triple of small diameter depends only on the geodesic distances between the points, then the manifold must be of constant curvature. This implies that if in a complete connected Riemannian manifold, the volume of the intersection of three small geodesic balls of equal radii depends only on the distances between the centers and the radius, then it is one of the simply connected spaces of constant curvature. This generalizes an earlier result of the first author and D. Kunszenti-Kovács (2010).

Regularity and uniqueness of p-harmonic maps with small range

We prove the uniqueness of solutions to Dirichlet problem for p-harmonic maps with images in a small geodesic ball of the target manifold. As a consequence, we show that such maps have Hölder continuous derivatives. This gives an extension of a result by Hildebrandt et al. (Acta Math 138:1–16, 1977) concerning harmonic maps.

A Characterization of harmonic spaces

The authors proved in "On the volume of the intersection of two geodesic balls," Differential Geom. Appl. 29 (2011), 567–576, that in a complete, connected, and simply connected Riemannian manifold, the volume of the intersection of two small geodesic balls depends only on the distance between the centers and the radii if and only if the space is harmonic. In this paper, we show that this proposition remains true, if the condition is restricted to balls of equal radii.

On the volume of the intersection of two geodesic balls

The first author and D. Kunszenti-Kovács (2010) [1] proved that if the volume of the intersection of three geodesic balls of a complete connected Riemannian manifold depends only on the center–center distances and the radii of the balls, then the manifold is one of the simply connected spaces of constant curvature. In this paper, we study the geometrical consequences of the analogous condition for pairs of geodesic balls. We show that in a complete, connected and simply connected Riemannian manifold, the volume of the intersection of two small geodesic balls depends only on the distance between the centers and the radii if and only if the space is harmonic. It is also shown that if in a Riemannian manifold the volume of the intersection of two small geodesic balls of equal radii depends only on the distance between the centers and the common value of the radii, then the space is Einstein, and if we assume in addition that the space is symmetric, then it must be Osserman and hence two-point homogeneous.

Heat Kernel Comparison on Alexandrov Spaces with Curvature Bounded Below

In this paper the comparison result for the heat kernel on Riemannian manifolds with lower Ricci curvature bound by Cheeger and Yau (1981) is extended to locally compact path metric spaces (X,d) with lower curvature bound in the sense of Alexandrov and with sufficiently fast asymptotic decay of the volume of small geodesic balls. As corollaries we recover Varadhan's short time asymptotic formula for the heat kernel (1967) and Cheng's eigenvalue comparison theorem (1975). Finally, we derive an integral inequality for the distance process of a Brownian Motion on (X,d) resembling earlier results in the smooth setting by Debiard, Geavau and Mazet (1975).

The Normalized mean Curvature Flow for a Small Bubble in a Riemannian Manifold

The evolution of an embedded surface under the normalized mean curvature flow is the result of a complicated interaction between the geometry of the evolving surface and the geometry of the ambient space, and is not well understood in the context of a general Riemannian manifold. In the present paper we identify a class of initial conditions, that we call "bubbles", whose dynamics is primarily determined by the ambient space. A bubble is an embedded surface that is close to a small geodesic ball; we find that its shape is robust along the evolution. Moreover, under a relatively tight condition relating shape to size, we show that the velocity of the center of the bubble is given, to principal order, by the gradient of the scalar curvature. Finally under natural conditions of compactness and nondegeneracy we show that such solutions converge, as t tends to infinity, to surfaces of constant mean curvature.

Jost maps, ball-homogeneous and harmonic manifolds

Given a real number ε>0, small enough, an associated Jost map J ε between two Riemannian manifolds is defined. Then we prove that connected Riemannian manifolds for which the center of mass of each small geodesic ball is the center of the ball (i.e. for which the identity is a J ε map) are ball-homogeneous. In the analytic case we characterize such manifolds in terms of the Euclidean Laplacian and we show that they have constant scalar curvature. Under some restriction on the Ricci curvature we prove that Riemannian analytic manifolds for which the center of mass of each small geodesic ball is the center of the ball are locally and weakly harmonic.

Mean distance of Brownian motion on a Riemannian manifold

Consider the mean distance of Brownian motion on Riemannian manifolds. We obtain the first three terms of the asymptotic expansion of the mean distance by means of stochastic differential equation for Brownian motion on Riemannian manifold. This method proves to be much simpler for further expansion than the methods developed by Liao and Zheng (Ann. Probab. 23(1) (1995) 173). Our expansion gives the same characterizations as the mean exit time from a small geodesic ball with regard to Euclidean space and the rank 1 symmetric spaces.

Yang–Mills fields and stochastic parallel transport in small geodesic balls

We develop a new method to obtain stochastic characterizations of Yang–Mills fields. Our main tool is the Itô-equation for the stochastic parallel transport. We estimate the drift terms in a small ball of radius ε and find that for a general connection the average rotation is of order ε 3 but that for a Yang–Mills connections the average rotation is of order ε 4. Using a Doob h-transform we give a new proof of the stochastic characterization of Yang–Mills fields by S. Stafford. Varying the starting point of the Brownian motion we obtain an unconditioned version of this result. By considering the horizontal Laplace equation we then apply our result to obtain a new analytic characterization of Yang–Mills fields.

Riemannian barycentres and geodesic convexity

Consider a closed subset of a complete Riemannian manifold, such that all geodesics with end-points in the subset are contained in the subset and the subset has boundary of codimension one. Is it the case that Riemannian barycentres of probability measures supported by the subset must also lie in the subset? It is shown that this is the case for 2-manifolds but not the case in higher dimensions: a counterexample is constructed which is a conformally-Euclidean 3-manifold, for which geodesics never self-intersect and indeed cannot turn by too much (so small geodesic balls satisfy a geodesic convexity condition), but is such that a probability measure concentrated on a single point has a barycentre at another point.

Semi-symmetric ball-homogeneous spaces and a volume conjecture

We prove that semi-symmetric ball-homogeneous spaces are locally symmetric and we use this result to prove that a semi-symmetric Riemannian manifold such that the volume of each sufficiently small geodesic ball is the same as in a Euclidean space, is locally flat.

Special Ball-Homogeneous Spaces

We continue the study of ball-homogeneous Riemannian manifolds, that is, R.ie-mannian spaces such that the volume of all sufficiently small geodesic balls or spheres only depends on the radius. First, we consider the case of locally reducible spaces. Then we treat the three-dimensional case, in particular for Einstein-like metrics and finally, we study confor-mally flat ball-homogeneous spaces. Our aim is to provide more partial answers to the question whether a ball-homogeneous space is locally homogeneous or not.

Wiener functionals associated with joint distributions of exit time and position from small geodesic balls

Consider the first exit time and position from small geodesic balls for Brownian motion on Riemannian manifolds. We establish a smooth Besselization technique and calculate the asymptotic expansion for the joint distributions by a purely probabilistic approach.