Related papers: A local optimal diastolic inequality on the two-sp…

In this paper, we prove an optimal isoperimetric inequality for spacelike, compact, star-shaped, and $2$-convex hypersurfaces in de Sitter space.

We prove two related results. The first is an ``Earthquake Theorem'' for closed hyperbolic surfaces with cone singularities where the total angle is less than $\pi$: any two such metrics in are connected by a unique left earthquake. The…

We prove a quantitative, large-scale doubling inequality and large-scale three-ellipsoid inequality for solutions of uniformly elliptic equations with periodic coefficients. These estimates are optimal in terms of the minimal length scale…

We give a concise proof that large classes of optimal (constant curvature or Einstein) pseudo-Riemannian metrics are maximally symmetric within their conformal class.

In this paper, we prove the strong Morse inequalities for the area functional in the space of embedded tori and spheres in the three sphere. As a consequence, we prove that in the three dimensional sphere with positive Ricci curvature,…

In a recent work, Galloway [9] proved a local foliation theorem by MOTSs for a 3-dimensional initial data set $(M,g,K)$ with mean curvature $\tau\le0$ in a 4-dimensional spacetime $(\overline M,\overline g)$ when (under suitable…

For a closed connected surface with a metric g, we consider the regularized trace of the inverse of the Laplace-Beltrami operator. We minimize this on the class of smooth metrics conformal to g having the same area, and show that the…

We consider the problem of finding embedded closed geodesics on the two-sphere with an incomplete metric defined outside a point. Various techniques including curve shortening methods are used.

We prove that the isometric embedding of any metric of differentiability class C1 in E3 exists. We use simplified notation for the given metric, namely geodesic parameters, and level parameters for the embedded surface in E3. Central to our…

In this paper, we prove the existence and uniqueness theorem for parabolic conical metrics on Riemann surfaces in the situation of generalized real angles, positive, zero and negative, by complex analysis, and give an example of this…

Consider the hyperplanes at a fixed distance $t$ from the center of the hypercube $[0,1]^d$. Significant attention has been given to determining the hyperplanes $H$ among these such that the $(d-1)$-dimensional volume of $H\cap[0,1]^d$ is…

Let $M$ be a closed hyperbolic 3-manifold that admits no infinitesimal conformally-flat deformations. Examples of such manifolds were constructed by Kapovich. Then if $g$ is a Riemannian metric on $M$ with scalar curvature greater than or…

We prove an inequality that generalizes the Fan-Taussky-Todd discrete analog of the Wirtinger inequality. It is equivalent to an estimate on the spectral gap of a weighted discrete Laplacian on the circle. The proof uses a geometric…

Gradients of the perimeter and area of a polygon have straightforward geometric interpretations. The use of optimality conditions for constrained problems and basic ideas in triangle geometry show that polygons with prescribed area…

We consider the isoperimetric inequality on the class of high-dimensional isotropic convex bodies. We establish quantitative connections between two well-known open problems related to this inequality, namely, the thin shell conjecture, and…

The paper is concerned with the maximization of Laplace eigenvalues on surfaces of given volume with a Riemannian metric in a fixed conformal class. A significant progress on this problem has been recently achieved by Nadirashvili-Sire and…

We continue our investigation of the space of geodesic laminations on a surface, endowed with the Hausdorff topology. We determine the topology of this space for the once-punctured torus and the 4-times-punctured sphere. For these two…

We describe an efficient algorithm to compute a conformally equivalent metric for a discrete surface, possibly with boundary, exhibiting prescribed Gaussian curvature at all interior vertices and prescribed geodesic curvature along the…

We describe all local Riemannian metrics on surfaces whose geodesic flows are superintegrable with one integral linear in momenta and one integral cubic in momenta. We also show that some of these metrics can be extended to the 2-sphere.…

Let $(M^n,g)$ be simply connected, complete, with non-positive sectional curvatures, and $\Sigma$ a 2-dimensional closed integral current (or flat chain mod 2) with compact support in $M$. Let $S$ be an area minimising integral 3-current…