Geodesic Balls Research Articles

The first Dirichlet eigenvalue and the width

For a geodesic ball with non-negative Ricci curvature and mean convex boundary, it is known that the first Dirichlet eigenvalue of this geodesic ball has a sharp lower bound in term of its radius. We show a quantitative explicit inequality, which bounds the width of geodesic ball in terms of the spectral gap between the first Dirichlet eigenvalue and the corresponding sharp lower bound.

Numerical optimization of Neumann eigenvalues of domains in the sphere

This paper deals with the numerical optimization of the first three eigenvalues of the Laplace-Beltrami operator of domains in the Euclidean sphere of R3 with Neumann boundary conditions. We address two approaches: the first one is a generalization of the initial problem leading to a density method and the other one is a shape optimization procedure using the level-set method. The original goal of those methods was to investigate the conjecture according to which the geodesic ball was optimal for the first non-trivial eigenvalue under certain conditions. These computations provide strong insight into the optimal shapes of those eigenvalue problems and show a rich variety of geometries regarding the proportion of the surface area of the sphere occupied by the domain. In the last part, the same algorithms are used to carry out the same investigations on a torus.

Asymptotic behavior of the first Dirichlet eigenvalue of AHE manifolds

AbstractIn this article, we investigate the rate at which the first Dirichlet eigenvalue of geodesic balls decreases as the radius approaches infinity. We prove that if the conformal infinity of an asymptotically hyperbolic Einstein manifold is of nonnegative Yamabe type, then the two‐term asymptotic of the eigenvalues is the same as that in hyperbolic space.

Llocp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p_{loc}$$\\end{document} Positivity Preservation and Liouville-Type Theorems

On a complete Riemannian manifold (M, g), we consider Llocp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{p}_{loc}$$\\end{document} distributional solutions of the differential inequality -Δu+λu≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-\\Delta u + \\lambda u \\ge 0$$\\end{document} with λ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda >0$$\\end{document} a locally bounded function that may decay to 0 at infinity. Under suitable growth conditions on the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{p}$$\\end{document} norm of u over geodesic balls, we obtain that any such solution must be nonnegative. This is a kind of generalized Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{p}$$\\end{document}-preservation property that can be read as a Liouville-type property for nonnegative subsolutiuons of the equation Δu≥λu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta u \\ge \\lambda u$$\\end{document}. An application of the analytic results to Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{p}$$\\end{document} growth estimates of the extrinsic distance of complete minimal submanifolds is also given.

Problem With Critical Sobolev Exponent and With Potential on SN

We consider the equation in D’, u = 0 on D′, where D′ is a geodesic ball with radius θ1, centered at the north pole, on , N ≥ 4, and q is a positive continuous function. We prove the existence of solutions that depends only on the behavior of the potential q near its minima.

Approaching the isoperimetric problem in HCm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{{\\mathbb {C}}}^m$$\\end{document} via the hyperbolic log-convex density conjecture

We prove that geodesic balls centered at some base point are uniquely isoperimetric sets in the real hyperbolic space HRn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{{\\mathbb {R}}}^n$$\\end{document} endowed with a smooth, radial, strictly log-convex density on the volume and perimeter. This is an analogue of the result by G. R. Chambers for log-convex densities on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^n$$\\end{document}. As an application we prove that in any rank one symmetric space of non-compact type, geodesic balls are uniquely isoperimetric in a class of sets enjoying a suitable notion of radial symmetry.

Liouville theorem for [formula omitted]-harmonic maps under non-negative [formula omitted]-Ricci curvature for non-positive [formula omitted

Let V be a C1-vector field on an n-dimensional complete Riemannian manifold (M,g). We prove a Liouville theorem for V-harmonic maps satisfying various growth conditions from complete Riemannian manifolds with non-negative (m,V)-Ricci curvature for m∈[−∞,0]∪[n,+∞] into Cartan–Hadamard manifolds, which extends Cheng’s Liouville theorem proved in Cheng (1980) for sublinear growth harmonic maps from complete Riemannian manifolds with non-negative Ricci curvature into Cartan–Hadamard manifolds. We also prove a Liouville theorem for V-harmonic maps from complete Riemannian manifolds with non-negative (m,V)-Ricci curvature for m∈[−∞,0]∪[n,+∞] into regular geodesic balls of Riemannian manifolds with positive upper sectional curvature bound, which extends the results of Hildebrandt et al. (1980) and Choi (1982). Our probabilistic proof of Liouville theorem for several growth V-harmonic maps into Hadamard manifolds enhances an incomplete argument in Stafford (1990). Our results extend the results due to Chen et al. (2012) and Qiu (2017) in the case of m=+∞ on the Liouville theorem for bounded V-harmonic maps from complete Riemannian manifolds with non-negative (∞,V)-Ricci curvature into regular geodesic balls of Riemannian manifolds with positive sectional curvature upper bound. Finally, we establish a connection between the Liouville property of V-harmonic maps and the recurrence property of ΔV-diffusion processes on manifolds. Our results are new even in the case V=∇f for f∈C2(M).

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.

Intersections of Poisson [formula omitted]-flats in constant curvature spaces

Poisson processes in the space of k-dimensional totally geodesic subspaces (k-flats) in a d-dimensional standard space of constant curvature κ∈{−1,0,1} are studied, whose distributions are invariant under the isometries of the space. We consider the intersection processes of order m together with their (d−m(d−k))-dimensional Hausdorff measure within a geodesic ball of radius r. Asymptotic normality for fixed r is shown as the intensity of the underlying Poisson process tends to infinity for all m satisfying d−m(d−k)≥0. For κ∈{−1,0} the problem is also approached in the set-up where the intensity is fixed and r tends to infinity. Again, if 2k≤d+1 a central limit theorem is shown for all possible values of m. However, while for κ=0 asymptotic normality still holds if 2k>d+1, we prove for κ=−1 convergence to a non-Gaussian infinitely divisible limit distribution in the special case m=1. The proof of asymptotic normality is based on the analysis of variances and general bounds available from the Malliavin–Stein method. We also show for general κ∈{−1,0,1} that, roughly speaking, the variances within a general observation window W are maximal if and only if W is a geodesic ball having the same volume as W. Along the way we derive a new integral-geometric formula of Blaschke–Petkantschin type in a standard space of constant curvature.

Saturation phenomena of a nonlocal eigenvalue problem: the Riemannian case

ABSTRACT In this paper we investigate the Riemannian extensibility of saturation phenomena treated first in the Euclidean framework by Brandolini et al. [Sharp estimates and saturation phenomena for a nonlocal eigenvalue problem. Adv Math (N Y). 2011;228(4):2352–2365.]. The saturation problem is formulated in terms of the first eigenvalue of the perturbation of the Laplace-Beltrami operator by the integral of the unknown function: the first eigenvalue increases with the weight affecting the integral up to a finite critical value and then remains constant, i.e. it saturates. Given a Riemannian manifold with certain curvature constraints, by using symmetrization arguments and sharp isoperimetric inequalities, we reduce the general problem to a variational one, formulated on either positively or negatively curved Riemannian model spaces; in addition, the possible scenarios for the optimal domains turn to be either geodesic balls or the union of two disjoint geodesic balls. We then explicitly compute the eigenvalues and eigenfunctions in terms of the radii, curvature and weight. A sufficient condition (incompatibility of a system of nonlinear equations involving special functions) is given that implies similar saturation phenomena to the Euclidean case. Due to its highly nonlinear character of the reduced problem (arising from the presence of curvature and special functions), we provide only partial answers to the original problem. However, both analytical computations and numerical tests suggest that the required incompatibility always persists. In addition, in the limit cases when the curvature tends to zero (for both positive an negative curvature), our results reduce to the Euclidean version.

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.

Free-boundary problems for holomorphic curves in the 6-sphere

We remark on two free-boundary problems for holomorphic curves in nearly-Kähler 6-manifolds. First, we observe that a holomorphic curve in a geodesic ball B of the round 6-sphere that meets $$\partial B$$ orthogonally must be totally geodesic. Consequently, we obtain rigidity results for reflection-invariant holomorphic curves in $$\mathbb {S}^6$$ and associative cones in $$\mathbb {R}^7$$ . Second, we consider holomorphic curves with boundary on a Lagrangian submanifold in a strict nearly-Kähler 6-manifold. By deriving a suitable second variation formula for area, we observe a topological lower bound on the Morse index. In both settings, our methods are complex-geometric, closely following arguments of Fraser–Schoen and Chen–Fraser.

Equilibria and energy minimizers for an interaction model on the hyperbolic space

We study an intrinsic model for collective behaviour on the hyperbolic space Hn. We investigate the equilibria of the aggregation equation (or equivalently, the critical points of the associated interaction energy) for interaction potentials that include Newtonian repulsion. By using the method of moving planes, we establish the radial symmetry and the monotonicity of equilibria supported on geodesic balls of Hn. We find several explicit forms of equilibria and show that one such equilibrium is a global energy minimizer. We also consider more general potentials and utilize a technique used for Rn to establish the existence of compactly supported global minimizers. Numerical simulations are presented, suggesting that some of the equilibria studied here are global attractors. The key tool in our investigations is a family of isometries of Hn that we have developed for this purpose.

On the volume functional of compact manifolds with harmonic Weyl tensor

AbstractThe main aim of this article is to give the complete classification of critical metrics of the volume functional on a compact manifold M with boundary and under the harmonic Weyl tensor condition. In particular, we prove that a critical metric with a harmonic Weyl tensor on a simply connected compact manifold with the boundary isometric to a standard sphere must be isometric to a geodesic ball in a simply connected space form , , and . To this end, we first conclude the classification of such critical metrics under the Bach‐flat assumption and then we prove that both geometric conditions are equivalent in this situation.

The construction of $$\epsilon $$-splitting map

For a geodesic ball with non-negative Ricci curvature and almost maximal volume, without using compactness argument, we construct an $$\epsilon $$ -splitting map on a concentric geodesic ball with uniformly small radius. There are two new technical points in our proof. The first one is the way of finding n directional points by induction and stratified almost Gou–Gu Theorem. The other one is the error estimates of projections, which guarantee the n directional points we find really determine n different directions.

A quantum complexity lower bound from differential geometry

The Bishop-Gromov bound -- a cousin of the focusing lemmas that Hawking and Penrose used to prove their black hole singularity theorems -- is a differential geometry result that upperbounds the rate of growth of volume of geodesic balls in terms of the Ricci curvature. In this paper, I apply the Bishop-Gromov bound to Nielsen's complexity geometry to prove lowerbounds on the quantum complexity of a typical unitary. For a broad class of penalty schedules, the typical complexity is shown to be exponentially large in the number of qubits. This technique gives results that are tighter than all known lowerbounds in the literature, as well as establishing lowerbounds for a much broader class of complexity geometry metrics than has hitherto been bounded. For some metrics, I prove these lowerbounds are tight. This method realizes the original vision of Nielsen, which was to apply the tools of differential geometry to study quantum complexity.

Linear isoperimetric inequalities for free boundary submanifolds in a geodesic ball

We derive linear isoperimetric inequalities for free boundary submanifolds in a geodesic ball of a Riemannian manifold in terms of the modified volume. It is known that the twice of the area of a free boundary minimal surface in a Euclidean unit ball is equal to the length of its boundary. This can be extended to space forms by using our linear isoperimetric inequalities for the modified volume. Moreover, we obtain a sharp lower bound for the modified volume of free boundary minimal surfaces in a geodesic ball of the upper hemisphere mathbb {S}_+^3. Finally, it is proved that the monotonicity property still holds for the modified volume of any submanifold in a complete simply connected Riemannian manifold with sectional curvature bounded above by a constant.

Two balls maximize the third Neumann eigenvalue in hyperbolic space

We show that the third eigenvalue of the Neumann Laplacian in hyperbolic space is maximal for the disjoint union of two geodesic balls, among domains of given volume. This extends a recent result by Bucur and Henrot in Euclidean space, while providing a new proof of a key step in their argument

Some sharp isoperimetric-type inequalities on Riemannian manifolds

We prove some sharp isoperimetric type inequalities for domains with smooth boundary on Riemannian manifolds. For example, using generalized convexity, we show that among all domains with a lower bound l for the cut distance and Ricci curvature lower bound (n−1)k, the geodesic ball of radius l in the space form of curvature k has the largest area-to-volume ratio. A similar but reversed inequality holds if we replace a lower bound on the cut distance by a lower bound of the mean curvature. As an application we show that C2 isoperimetric domains in standard space forms are balls. We also prove another isoperimetric inequality involving the extrinsic radius of a domain when the curvature of the ambient space is bounded above. We then extend it to an inequality which involves the Hausdorff measure of the cut locus.

On the $$\sigma _2$$-curvature and volume of compact manifolds

In this work, we are interested in studying deformations of the \(\sigma _2\)-curvature and the volume. For closed manifolds, we relate critical points of the total \(\sigma _2\)-curvature functional to the \(\sigma _2\)-Einstein metrics and, as a consequence of results of Gursky and Viaclovsky (Invent Math 145(2):251–278, 2001) and Hu and Li (Trans Am Math Soc 356(8):3005–3023, 2004), we obtain a sufficient and necessary condition for a critical metric to be Einstein. Moreover, we show a volume comparison result for Einstein manifolds with respect to \(\sigma _2\)-curvature which shows that the volume can be controlled by the \(\sigma _2\)-curvature under certain conditions. Next, for compact manifold with nonempty boundary, we study variational properties of the volume functional restricted to the space of metrics with constant \(\sigma _2\)-curvature and with fixed induced metric on the boundary. We characterize the critical points to this functional as the solutions of an equation and show that in space forms they are geodesic balls. Studying second-order properties of the volume functional, we show that there is a variation for which geodesic balls are indeed local minima in a natural direction.