arXiv research

The paper characterizes unit balls among Stein spaces with specific groups using Bergman-Einstein metrics.

We determine the Christoffel's symbols for the Siegel-Jacobi ball endowed with the balanced metric. We study the equations of geodesics on the Siegel-Jacobi ball. We calculate the covariant derivative of one-forms in the variables in which is expressed the balanced metric on the Siegel-Jacobi ball.

In a compact orbifold, for small prescribed volume, an isoperimetric region is close to a small metric ball; in a Euclidean orbifold, it is a small metric ball.

Study minimal networks on spheres and balls near standard metrics.

Study on when Bergman metrics of domains are induced by balls.

Study free boundary minimal submanifolds in geodesic balls in hyperbolic and spherical spaces.

Paper finds critical metrics with pinched curvature are geodesic balls.

No radial balanced metrics found on Kepler manifold unit ball with mild boundary conditions.

New method to bound Laplacian eigenvalues of geodesic balls.

Classifies metrics with specific curvature properties on a ball.

We determine the matrix of the balanced metric of the Siegel-Jacobi ball and its inverse. We calculate the scalar curvature, the Ricci form and the Laplace-Beltrami operator of this manifold. We discuss several geometric aspects related with Berezin quantization on the Siegel-Jacobi ball.

We exhibit an explicit one-parameter smooth family of Poincaré-Einstein metrics on the even-dimensional unit ball whose conformal infinities are the Berger spheres. Our construction is based on a Gibbons-Hawking-type ansätz of Page and Pope. The family contains the hyperbolic metric, converges to the complex hyperbolic…

In this paper, we construct an asymptotically hyperbolic metric with scalar curvature -6 on unit ball $\mathbf{D}^3$, which contains multiple horizons.

Paper proves uniqueness of Einstein metrics on balls.

In this note we provide natural optimal geometric conditions for a Riemannian manifold suitably covered by two open metric balls to be homeomorphic to a sphere. This can be viewed as a geometric analogue of Brown's theorem in topology stating that a closed manifold covered by two topological balls is a sphere.

Local isoperimetric inequality holds for balls with nonpositive curvature.

Study shows volume constraints lead to isoperimetric constant bounds in specific metric spaces.

Study of Riemannian geometry on quaternionic unit ball linked to Sp(1,1) group.

We prove explicit upper and lower bounds for the $L^1$-moment spectra for the Brownian motion exit time from extrinsic metric balls of submanifolds $P^m$ in ambient Riemannian spaces $N^{n}$. We assume that $P$ and $N$ both have controlled radial curvatures (mean curvature and sectional curvature, respectively) as view…

We prove that the Teichmüller space of surfaces of genus $\mathbf{g}$ with $\mathbf{p}$ punctures contains balls which are not convex in the Teichmüller metric whenever $3\mathbf{g}-3+\mathbf{p} > 1$.

The study finds at least 2 free-boundary minimal disks in convex 3-balls.

We construct non-trivial continuous isospectral deformations of Riemannian metrics on the ball and on the sphere in $\R^n$ for every $n\geq 9$. The metrics on the sphere can be chosen arbitrarily close to the round metric; in particular, they can be chosen to be positively curved. The metrics on the ball are both Diric…

This paper considers the prescribed zero scalar curvature and mean curvature problem on the n-dimensional Euclidean ball for $n \geq 3$. Given a rotationally symmetric function $H:\partial B^{n}\rightarrow R$, in this work, we will prove that if $H'(r)$ changes signs where $H>0$ and $H(r)$ also satisfies a flatness con…

New functionals defined for free boundary minimal submanifolds in higher dimensions.

Two-dimensional domains with Kähler-Einstein Bergman metrics are biholomorphic to the unit ball.

Study on curvature flow in 4D ball, proving existence and convergence.

We show that metrics that maximize the k-th Steklov eigenvalue on surfaces with boundary arise from free boundary minimal surfaces in the unit ball. We prove several properties of the volumes of these minimal submanifolds. For free boundary minimal submanifolds in the ball we show that the boundary volume is reduced up…

We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In…

We construct pairs of conformally equivalent isospectral Riemannian metrics $φ_1 g$ and $φ_2 g$ on spheres $S^n$ and balls $B^{n+1}$ for certain dimensions $n$, the smallest of which is $n=7$, and on certain compact simple Lie groups. In the case of Lie groups, the metric $g$ is left-invariant. In the case of spheres a…

Take a torus with a Riemannian metric. Lift the metric on its universal cover. You get a distance which in turn yields balls. On these balls you can look at the Laplacian. Focus on the spectrum for the Dirichlet or Neumann problem. We describe the asymptotic behaviour of the eigenvalues as the radius of the balls goes …

Simply connected 4-manifolds with specific Weyl tensor are geodesic balls in space forms.

We prove that the space of smooth Riemannian metrics on the three-ball with non-negative Ricci curvature and strictly convex boundary is path connected; and, moreover, that the associated moduli space (i.e., modulo orientation-preserving diffeomorphisms of the three-ball) is contractible. As an application, using resul…

A conformal metric on a 4-ball induces on the boundary 3-sphere a conformal metric and a trace-free second fundamental form. Conversely, such a data on the 3-sphere is the boundary of a unique selfdual conformal metric, defined in a neighborhood of the sphere. In this paper we characterize the conformal metrics and tra…

Uniform comparison of hyperbolic ball volumes on universal cover.

The paper proves the stability of a 3-ball under curvature constraints.

Using the flow method, we prove some existence results for the problem of prescribing the mean curvature on the unit ball. More precisely, we prove that there exists a conformal metric on the unit ball such that its mean curvature is $f$, when $f$ possesses certain reflection or rotation symmetry.

Geometric structures on quaternionic unit ball for slice regular Möbius transformations.

We obtain lower bounds for the first Laplacian eigenvalues of geodesic balls of spherically symmetric manifolds. These lower bounds are only $C^{0}$ dependent on the metric coefficients.

David Hilbert discovered in 1895 an important metric that is canonically associated to any convex domain $Ω$ in the Euclidean (or projective) space. This metric is known to be Finslerian, and the usual proof assumes a certain degree of smoothness of the boundary of $Ω$ and refers to a theorem by Busemann and Mayer that…

We equip the whole tangent space $TM$ to a hyperbolic manifold $M$ (of constant sectional curvature -1) with a natural metric in an intrinsic way, so that the isometries of $M$ extend to isometries of $TM$ by holomorphic continuation. The image to the tangent space to a geodesic is equivalent to a hyperbolic disk. In t…

The unit ball is characterized by a Kähler-Einstein potential.

We consider the rate of volume growth of large Carnot-Carathéodory metric balls on a class of unbounded model hypersurfaces in $\mathbb{C}^2$. When the hypersurface has a uniform global structure, we show that a metric ball of radius $δ\gg 1$ either has volume on the order of $δ^3$ or $δ^4$. We also give necessary and …

Let (M,g) be a compact n-dimensional Riemannian manifold with boundary. This article is concerned with the set of scalar-flat metrics on M which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. We construct examples of metrics on the unit ball, in dimensions n>=25, for wh…

In this paper we study a boundary value problem for the Ricci flow in the two dimensional ball endowed with a rotationally symmetric metric. We show short and long time existence results. We construct families of metrics for which the flow uniformizes the curvature along a sequence of times. Finally we show that if the…

Study $σ_2$-curvature and volume of compact manifolds, proving conditions for Einstein metrics and geodesic balls.

New metric properties show volume constraints in collapsing spaces.

For a Riemannian polyhedra, we study the geometry of the unit ball for the unidimensional stable norm (stable ball). In the case of a unidimensional Riemannian polyhedra (graph), we show that the stable ball is a polytope whose vertices are completely described by combinatorial properties of the graph. We study then th…

The smallest $r$ so that a metric $r$-ball covers a metric space $M$ is called the radius of $M$. The volume of a metric $r$-ball in the space form of constant curvature $k$ is an upper bound for the volume of any Riemannian manifold with sectional curvature $\geq k$ and radius $\leq r$. We show that when such a manifo…