arXiv research

The paper develops inequalities for log-concave functions and related surface areas.

In this paper, we introduce the $L_p$ geominimal surface area for all $-n\neq p<1$, which extends the classical geominimal surface area ($p=1$) by Petty and the $L_p$ geominimal surface area by Lutwak ($p>1$). Our extension of the $L_p$ geominimal surface area is motivated by recent work on the extension of the $L_p$ a…

Proves strong Morse inequalities for area functional in low dimensions.

In this paper, we introduce several mixed $L_p$ geominimal surface areas for multiple convex bodies for all $p\neq -n$. Our definitions are motivated from an equivalent formula for the mixed $p$-affine surface area. Some properties, such as the affine invariance, for these mixed $L_p$ geominimal surface areas are prove…

New optimal isosystolic inequality found for Finsler reversible 2-tori.

The article disproves a local systolic inequality and shows a lower bound on filling area.

Study spherical convex bodies using $L_p$-floating areas and curvature entropy.

Area-charge inequalities and local rigidity of free boundary MOTS in charged initial data sets

The paper proves inequalities for hyperbolic sets and curves.

Paper proves new inequalities for Einstein-Maxwell data sets.

New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.

New inequality linking geodesic length and volume in complex projective plane.

The paper proves a diastolic inequality linking surface area and loop length.

We investigate the filling area conjecture, optimal systolic inequalities, and the related problem of the nonvanishing of certain linking numbers in 3-manifolds.

The paper proves optimal systolic inequalities for Möbius strip and Klein bottle.

This paper aims to develop basic theory for the dual Orlicz $L_φ$ affine and geominimal surface areas for star bodies, which belong to the recent dual Orlicz-Brunn-Minkowski theory for star bodies. Basic properties for these new affine invariants will be provided. Moreover, related Orlicz affine isoperimetric inequalit…

The study proves analogues of the discrete isoperimetric inequality in hyperbolic geometry.

We establish the conjectured area-angular momentum-charge inequality for stable apparent horizons in the presence of a positive cosmological constant, and show that it is saturated precisely for extreme Kerr-Newman-de Sitter horizons. As with previous inequalities of this type, the proof is reduced to minimizing an `ar…

The paper explores rigid geometric structures near surfaces with equality in area-charge inequalities.

The discrete isoperimetric inequality in Euclidean geometry states that among all $n$-gons having a fixed perimeter $p$, the one with the largest area is the regular $n$-gon. The statement is true in spherical geometry and hyperbolic geometry as well. In this paper, we generalize the discrete isoperimetric inequality t…

The Riemannian Penrose inequality (RPI) bounds from below the ADM mass of asymptotically flat manifolds of nonnegative scalar curvature in terms of the total area of all outermost compact minimal surfaces. The general form of the RPI is currently known for manifolds of dimension up to seven. In the present work, we pro…

Minimal surfaces in a ball have limited area.

We prove two weighted geometric inequalities that hold for strictly mean convex and star-shaped hypersurfaces in Euclidean space. The first one involves the weighted area and the area of the hypersurface and also the volume of the region enclosed by the hypersurface. The second one involves the total weighted mean curv…

A new systolic inequality with a remainder for the real projective plane.

Paper proves Harnack inequality for $f$-mean curvature flow.

Gradient flow method solves isoperimetric inequality for maps.

In this paper, we deals with isoperimetric-type inequalities for closed convex curves in the Euclidean plane R^2. We derive a family of parametric inequalities involving the following geometric functionals associated to a given convex curve with a simple Fourier series proof: length, area of the region included by the …

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.

We note an area-charge inequality orignially due to Gibbons: if the outermost horizon $S$ in an asymptotically flat electrovacuum initial data set is connected then $|q|\leq r$, where $q$ is the total charge and $r=\sqrt{A/4π}$ is the area radius of $S$. A consequence of this inequality is that for connected black hole…

Study on $L_p$ affine surface areas and their inequalities for convex bodies.

Improved bounds on curve filling areas in Banach spaces, leading to rigidity of Pu's inequality.

The paper extends geometric inequalities from Euclidean space to Riemannian manifolds.

Michael-Simon inequality proven for anisotropic energies close to area.

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

Lipschitz maps on metric surfaces are rigid if they preserve area.

Proves mass-capacity inequalities for critical area-normalized capacitors, improving Schwarzschild metric uniqueness.

Using a ramified cover of the two-sphere by the torus, we prove a local optimal inequality between the diastole and the area on the two-sphere near a singular metric. This singular metric, made of two equilateral triangles glued along their boundary, has been conjectured by E. Calabi to achieve the best ratio area over…

Sharp inequalities for curved surfaces and cones.

Schwarzschild 3-manifold stability proven for 3D Penrose inequality.

Proves a conjecture about metrics and minimal area enclosures.

Affine $λ$-equidistants of convex polygons with parallel opposite sides have applications to isoperimetric inequalities.

The classical isoperimetric inequality in the Euclidean plane $\mathbb{R}^2$ states that for a simple closed curve $M$ of the length $L_{M}$, enclosing a region of the area $A_{M}$, one gets \begin{align*} L_{M}^2\geqslant 4πA_{M}. \end{align*} In this paper we present the improved isoperimetric inequality, which state…

Sharp inequality for Lorentzian spaces with timelike Ricci bounds.

Minimal surfaces in hyperbolic space have a sharp area bound.

We introduce notions of Cheeger constants for graphons and graphings. We prove Cheeger and Buser inequalities for these. On the way we prove co-area formulae for graphons and graphings.

The study proves inequalities for area and boundary length of disks in convex manifolds.

Study flat flow solutions to Mullins-Sekerka and area-preserving curvature flows on planar flat torus.

We show that a smooth unknotted curve in R^3 satisfies an isoperimetric inequality that bounds the area of an embedded disk spanning the curve in terms of two parameters: the length L of the curve and the thickness r (maximal radius of an embedded tubular neighborhood) of the curve. For fixed length, the expression giv…