arXiv research

Groups satisfy linear surface isoperimetric functions.

We show that any star-shaped convex hypersurface with constant Weingarten curvature in the deSitter-Schwarzschild manifold is a sphere of symmetry. Moreover, we study an isoperimetric problem for bounded domains in the doubled Schwarzschild manifold. We prove the existence of an isoperimetric surface for any value of t…

Smooth minimizers found for Willmore energy surfaces.

Uniform Lipschitz continuity of isoperimetric profiles in evolving surfaces.

Optimal constants for isoperimetric inequalities involving Steklov eigenvalues on surfaces are determined.

Flow preserves isoperimetric ratio for immersed surfaces.

The isoperimetric ratio of an embedded surface in $R^3$ is defined as the ratio of the area of the surface to power three to the squared enclosed volume. The aim of the present work is to study the minimization of the Willmore energy under fixed isoperimetric ratio when the underlying abstract surface has fixed genus $…

The paper proves a reverse isoperimetric inequality and applies it to analyze surface flows.

Researchers find a surface with minimum bending energy for any genus and isoperimetric ratio.

We show that the Cheeger constant of compact surfaces is bounded by a function of the area. We apply this to isoperimetric profiles of bounded genus non-compact surfaces, to show that if their isoperimetric profile grows faster than $\sqrt t$, then it grows at least as fast as a linear function. This generalizes a resu…

Sharp inequalities and symmetries on Riemannian surfaces quantified.

Study on isoperimetric inequalities and regularity of $A$-harmonic functions on surfaces.

Prove an isoperimetric inequality for compact bodies in 3D contact non-unimodular Lie groups.

Minimal surfaces in hyperbolic space have a sharp area bound.

We solve the isoperimetric problem in the Lens spaces with large fundamental group. Namely, we prove that the isoperimetric surfaces are geodesic spheres or tori of revolution about geodesics. We also show that the isoperimetric problem in L(3,1) and L(3,2) follows from the proof of the Willmore conjecture by Marques a…

New proof of surface group theorem for 2D Poincaré duality groups.

For general varifolds in Euclidean space, we prove an isoperimetric inequality, adapt the basic theory of generalised weakly differentiable functions, and obtain several Sobolev type inequalities. We thereby intend to facilitate the use of varifold theory in the study of diffused surfaces.

We show that there is a complete connected 2-dimensional Riemannian manifold with discontinuous isoperimetric profile, answering a question of Nardulli and Pansu.

In the class of smoothly embedded surfaces of sphere type we prove that the isoperimetric deficit can be controlled by the Willmore deficit.

Sharp inequalities for curved surfaces and cones.

Paper proves flat 3-manifolds with positive mass have unique isoperimetric surfaces.

Formula calculates volume of CMC surfaces with translational periods, disproving isoperimetric conjecture.

Sharp Minkowski inequality for convex surfaces in curved spaces.

We establish some a priori geometric relations on stable minimal surfaces lying inside three-manifolds with scalar curvature uniformly bounded below. The relations are based on a slight generalization of a formula due to Castillon. We apply it to prove non-local rigidity results in the particular sense that they expres…

The paper solves reverse isoperimetric problems for convex bodies with curvature constraints.

The article proves Randers Poincaré disc satisfies isoperimetric equality.

Paper proves constants for Moser-Trudinger inequality on surfaces.

Study sharp upper bounds for Aharonov-Bohm eigenvalues on surfaces.

We equip many non compact non simply connected surfaces with smooth Riemannian metrics whose isoperimetric profile is smooth, a highly non generic property. The computation of the profile is based on a calibration argument, a rearrangement argument, the Bol-Fiala curvature dependent inequality, together with new result…

We review and extend here some recent results on the existence of minimal surfaces and isoperimetric sets in non homogeneous and anisotropic periodic media. We also describe the qualitative properties of the homogenized surface tension, also known as stable norm (or minimal action) in Weak KAM theory. In particular we …

Long time existence and convergence to a circle is proved for radial graph solutions to a mean curvature type curve flow in warped product surfaces (under a weak assumption on the warp potential of the surface). This curvature flow preserves the area enclosed by the evolving curve, and this fact is used to prove a gene…

In this paper we establish a general form of the isoperimetric inequality for immersed closed curves (possibly non-convex) in the plane under rotational symmetry. As an application we obtain a global existence result for the surface diffusion flow, providing that an initial curve is $H^2$-close to a multiply covered ci…

The study solves the isoperimetric problem for Heisenberg group norms.

We study the topology of (properly) immersed complete minimal surfaces $P^2$ in Hyperbolic and Euclidean spaces which have finite total extrinsic curvature, using some isoperimetric inequalities satisfied by the extrinsic balls in these surfaces, (see \cite{Pa}). We present an alternative and partially unified proof of…

Study on surfaces in Heisenberg group with constant mean curvature.

We characterize the standard $\mathbb{S}^3$ as the closed Ricci-positive 3-manifold with scalar curvature at least 6 having isoperimetric surfaces of largest area: $4π$. As a corollary we answer in the affirmative an interesting special case of a conjecture of Min-Oo's on the scalar curvature rigidity of the upper hemi…

Study minimizers in large volume isoperimetric problems with a new flatness criterion.

Study on lengths and curvatures of harmonic functions on smooth and singular surfaces.

We give a brief literature review of the isoperimetric problem and discuss its relationship with the Cheeger constant of Riemannian $n$-manifolds. For some non-compact, finite area 2-manifolds, we prove the existence and regularity of subsets whose isoperimetric ratio is equal to the Cheeger constant. To do this, we us…

A comparison theorem for the isoperimetric profile on the universal cover of surfaces evolving by normalised Ricci flow is proven. For any initial metric, a model comparison is constructed that initially lies below the profile of the initial metric and which converges to the profile of the constant curvature metric. Th…

The renormalized volume is reinterpreted using isoperimetric profiles.

In 3D space forms, a lens minimizes volume for a fixed surface area.

Proves strict inequality for minimizers of Willmore energy under isoperimetric constraints.

New illumination bodies defined for ball-convex shapes, proving convexity and establishing surface area measures.

Two families of general affine surface areas are introduced. Basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for $L_φ$ affine surface areas are established.

Study eigenvalues of Dirac operator on surfaces, proving existence and deriving inequalities.

We prove an Hersch's type isoperimetric inequality for the third positive eigenvalue on $\mathbb S^2$. Our method builds on the theory we developped to construct extremal metrics on Riemannian surfaces in conformal classes for any eigenvalue.

In this paper we first prove some linear isoperimetric inequalities for submanifolds in the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds. Moreover, the equality is attained. Next, we prove some monotonicity formulas for submanifolds with bounded mean curvature vector in warped product manifolds and, as cons…