arXiv research

We study the classification of area-stationary and stable $C^2$ regular surfaces in the space of the rigid motions of the Minkowski plane E(1,1), equipped with its sub-Riemannian structure. We construct examples of area-stationary surfaces that are not foliated by sub-Riemannian geodesics. We also prove that there exis…

A bound on surface area in Riemannian manifolds without totally geodesic surfaces.

The paper proves rigidity theorems for area widths of Riemannian manifolds.

The study finds large area minimal surfaces in certain manifolds.

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

We show the existence of a smooth spherical surface minimizing the Willmore functional subject to an area constraint in a compact Riemannian three-manifold, provided the area is small enough. Moreover, we classify complete surfaces of Willmore type with positive mean curvature in Riemannian three-manifolds.

Study shows area-minimizing submanifolds are mostly rough, not smooth.

New minimal surfaces can have huge area and index.

Study estimates hypersurface areas in curved spaces, with applications to spectrum bounds.

The book explores stochastic areas and heat kernels on manifolds.

In Riemannian manifolds, minimal hypersurfaces with large area exist or have complex structures.

The Riemannian hemisphere has a lower bound for its mass.

New metric creates a surface with infinite topology.

The study finds instability conditions for specific surfaces in sub-Riemannian 3-space forms.

A sphere has at least two geodesics whose product length is bounded by a constant times the area.

Lower bounds for surface area and volume of convex hypersurfaces.

The paper studies Pansu spheres in a sub-Riemannian 3-sphere and their area-minimizing properties.

We prove flatness of complete Riemannian planes and cylinders without conjugate points under optimal conditions on the area growth.

Proves a conjecture about metrics and minimal area enclosures.

Estimate sphere area in Sol group up to a factor of 10.

Extends illumination bodies to non-Euclidean spaces and proves their volume derivative defines surface area.

Extended systolic inequality for 2-complexes to improve group systolic area bounds.

We consider surfaces of class $C^1$ in the $3$-dimensional sub-Riemannian Heisenberg group ${\mathbb H}^1$. Assuming the surface is area-stationary, i.e., a critical point of the sub-Riemannian perimeter under compactly supported variations, we show that its regular part is foliated by horizontal straight lines. In cas…

Quantizes Willmore energy in Riemannian manifolds with bounded energy and area.

Let $(M,g)$ be a 3-dimensional Riemannian manifold. The goal of the paper it to show that if $P_{0}\in M$ is a non-degenerate critical point of the scalar curvature, then a neighborhood of $P_{0}$ is foliated by area-constrained Willmore spheres. Such a foliation is unique among foliations by area-constrained Willmore …

We construct embedded Willmore tori with small area constraint in Riemannian three-manifolds under some curvature condition used to prevent Möbius degeneration. The construction relies on a Lyapunov-Schmidt reduction; to this aim we establish new geometric expansions of exponentiated small symmetric Clifford tori and a…

K-area is an invariant for Riemannian manifolds introduced by Gromov as an obstruction to the existence of positive scalar curvature. However in general it is difficult to determine whether K-area is finite or not. though the definition of K-area is quite natural. In this paper, we study how the invariant changes under…

Proves upper bound on systolic ratio for circle fillings.

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

The paper solves the isoperimetric problem in Riemannian optical geometry, proving circles minimize lengths with area constraints.

Let $M$ be a weighted manifold with boundary $\partial M$, i.e., a Riemannian manifold where a density function is used to weight the Riemannian Hausdorff measures. In this paper we compute the first and the second variational formulas of the interior weighted area for deformations by hypersurfaces with boundary in $\p…

Let $M$ be a complete Sasakian sub-Riemannian $3$-manifold of constant Webster scalar curvature $κ$. For any point $p\in M$ and any number $λ\in\mathbb{R}$ with $λ^2+κ>0$, we show existence of a $C^2$ spherical surface $\mathcal{S}_λ(p)$ immersed in $M$ with constant mean curvature $λ$. Our construction recovers in par…

P. Papasoglu asked in [Pap13] whether for any Riemannian 3-disk $M$ with diameter $d$, boundary area $A$ and volume $V$, there exists a homotopy $S_t$ contracting the boundary to a point so that the area of $S_t$ is bounded by $f(d,A,V)$ for some function $f$. He further asks whether it is possible to subdivide $M$ by …

This paper tackles Gromov's filling area conjecture using discrete graph theory.

We prove that simply connected open Riemannian manifolds of bounded geometry, linear growth and sublinear filling growth (e.g. finite filling area) are simply connected at infinity.

Study proves area estimates for stable capillary hypersurfaces with nonpositive Yamabe invariant.

In this paper we prove the following. Let $Σ$ be an $n$--dimensional closed hyperbolic manifold and let $g$ be a Riemannian metric on $Σ\times \mathbb{S}^1$. Given an upper bound on the volumes of unit balls in the Riemannian universal cover $(\widetilde{Σ\times \mathbb{S}^1},\widetilde{g})$, we get a lower bound on th…

We show that in the first sub-Riemannian Heisenberg group there are intrinsic graphs of smooth functions that are both critical and stable points of the sub-Riemannian perimeter under compactly supported variations of contact diffeomorphisms, despite the fact that they are not area-minimizing surfaces. In particular, w…

The study examines the graphical mean curvature flow on compact manifolds with bounded bi-Ricci curvature.

Constructs area-minimizing submanifolds with fractal singularities.

Study on geometry of mountain slopes as Finsler metrics.

We show that an asymptotically flat Riemannian three-manifold with non-negative scalar curvature is isometric to flat $\mathbb{R}^3$ if it admits an unbounded area-minimizing surface. This answers a question of R. Schoen.

This is the last of a series of three papers in which we give a new, shorter proof of a slightly improved version of Almgren's partial regularity of area minimizing currents in Riemannian manifolds. Here we perform a blow-up analysis deducing the regularity of area minimizing currents from that of Dir-minimizing multip…

We show that if $F$ is a smooth, closed, orientable surface embedded in a closed, orientable 3-manifold $M$ such that for each Riemannian metric $g$ on $M$, $F$ is isotopic to a least-area surface $F(g)$, then $F$ is incompressible.

This is the second of a series of two papers where we construct embedded Willmore tori with small area constraint in Riemannian three-manifolds. In both papers the construction relies on a Lyapunov-Schmidt reduction, the difficulty being the Möbius degeneration of the tori. In the first paper the construction was perfo…

We develop a bubble tree construction and prove compactness results for $W^{2,2}$ branched conformal immersions of closed Riemann surfaces, with varying conformal structures whose limit may degenerate, in a compact Riemannian manifold with uniformly bounded areas and Willmore energies. The compactness property is appli…

It is a classical theorem of Loewner that the systole of a Riemannian torus can be bounded in terms of its area. We answer a question of a similar flavor of Robert Young showing that if $T$ is a Riemannian 2-torus with boundary in $\mathbb R ^n$, such that the boundary curve is a standard unit circle, then the length o…

Max systoles on spheres with punctures are counted.