Related papers: Volume bounds of conic 2-spheres
In this note we show the following result using the integral-geometric formula of R. Howard: Consider the totally geodesic $\mathbb{R}P^{2m}$ in $\mathbb{C}P^n$. Then it minimizes volume among the isotropic submanifolds in the same…
We study the infimum of the renormalized volume for convex-cocompact hyperbolic manifolds, as well as describing how a sequence converging to such values behaves. In particular, we show that the renormalized volume is continuous under the…
We derive quantitative volume constraints for sampling measures $\mu_t$ on the unit sphere $\mathbb{S}^d$ that satisfy Marcinkiewicz-Zygmund inequalities of order $t$. Using precise localization estimates for Jacobi polynomials, we obtain…
In this article is given a simple expression for the \textit{ center of mass} for a system of material points in a two-dimensional surface of constant negative Gaussian curvature. Using basic techniques of Geometry, an expression in…
We discuss infinitesimal isometries of the middle surfaces and present some characteristic conditions for a function to be the normal component of an infinitesimal isometry. Our results show that those characteristic conditions depend on…
We define a renormalized volume for a region in an asymptotically hyperbolic Einstein manifold that is bounded by a Graham-Witten minimal surface and the conformal infinity. We prove a Gauss-Bonnet theorem for the renormalized volume, and…
We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body $C\subset \mathbb{R}^{n+1}$, without assuming any further regularity on the boundary of $C$. Motivated by an example of an…
We give an explicit estimate of the distance of a closed, connected, oriented and immersed hypersurface of a space form to a geodesic sphere and show that the spherical closeness can be controlled by a power of an integral norm of the…
We obtain new topological information about the local structure of collapsing under a lower sectional curvature bound. As an application we prove a new sphere theorem and obtain a partial result towards the conjecture that not every…
Gromov and Piatetski-Shapiro proved existence of finite volume non-arithmetic hyperbolic manifolds of any given dimension. In dimension four and higher, we show that there are about v^v such manifolds of volume at most v, considered up to…
We study Gromov's problem concerning minimal normal curvature immersions in the unit ball. In particular, we determine the minimal possible value of the normal curvature of an $S^n\times S^1$. We also prove a differentiable sphere theorem…
We establish a min-max estimate on the volume width of a closed Riemannian manifold with nonnegative Ricci curvature. More precisely, we show that every closed Riemannian manifold with nonnegative Ricci curvature admits a PL Morse function…
We construct an explicit lower bound for the volume of a complex hyperbolic orbifold that depends only on dimension.
We give a lower bound for the Gaussian curvature of convex level sets of minimal graphs and the solutions to semilinear elliptic equations with the norm of boundary gradient and the Gaussian curvature of the boundary.
We study the size estimate problem for the two-phase shallow shell equations in this paper. Our aim is to derive bounds on the volume fraction of each phase assuming that the material properties of the two phases are given. The approach in…
We investigate the problem of finding complete strictly convex hypersurfaces of constant curvature in hyperbolic space with a prescribed asymptotic boundary at infinity for a general class of curvature functions.
The Gromov-Hausdorff distance between two metric spaces measures how far the spaces are from being isometric. It has played an important and longstanding role in geometry and shape comparison. More recently, it has been discovered that the…
In this paper we provide a geometric condition satisfied by certain closed subsets of the Riemann sphere which implies that their hyperbolic convex hulls in $\mathbb{H}^3$ have infinite volume. As a corollary, we characterize continua in…
The Hessian of the renormalized volume of geometrically finite hyperbolic $3$-manifolds without rank-$1$ cusps, computed at the hyperbolic metric $g$ with totally geodesic boundary of the convex core, is shown to be a strictly positive…
We define a capacity which measures the size of Weinstein tubular neighbourhoods of Lagrangian submanifolds. In symplectic vector spaces this leads to bounds on the codisc radius for any closed Lagrangian submanifold in terms of Viterbo's…