arXiv research

Ecker's and Huisken's quantities agree for ancient mean curvature flows.

The flag curvature of a Finsler metric is called a Riemannian quantity because it is an extension of sectional curvature in Riemannian geometry. In Finsler geometry, there are several non-Riemannian quantities such as the (mean) Cartan torsion, the (mean) Landsberg curvature and the S-curvature, which all vanish for Ri…

Derive monotone quantities for harmonic functions on asymptotically flat 3-manifolds with nonnegative scalar curvature.

New quantity helps map homotopy classes in complex spaces.

We give a family of monotone quantities along smooth solutions to the inverse curvature flows in Euclidean spaces. We also derive a related geometric inequality for closed hypersurfaces with positive k-th mean curvature.

We prove that strictly convex surfaces moving by $K^{α/2}$ become spherical as they contract to points, provided $α$ lies in the range $[1,2]$. In the process we provide a natural candidate for a curvature pinching quantity for surfaces moving by arbitrary functions of curvature, by finding a quantity conserved by the …

Derives monotonic quantities for $p$-harmonic functions on manifolds.

This survey article is about discrete constant mean curvature surfaces defined by an approach related to integrable systems techniques. We introduce the notion of discrete constant mean curvature surfaces by first introducing properties of smooth constant mean curvature surfaces. We describe the mathematical structure …

We consider flows with normal velocities equal to powers strictly larger than one of the Gauss curvature. Under such flows closed strictly convex surfaces converge to points. In his work on the square of the norm of the second fundamental form, Schnürer proposes criteria for selecting quantities that are suitable for p…

Study almost rigidity of super Ricci flow with non-negative Muller quantity.

In this paper we study smooth solutions to a fractional mean curvature flow equation. We establish a comparison principle and consequences such as uniqueness and finite extinction time for compact solutions. We also establish evolutions equations for fractional geometric quantities that yield preservation of certain qu…

We find a new monotone increasing quantity along smooth solutions to the inverse mean curvature flow in $\mathbb R^n$. As an application, we derive a sharp geometric inequality for mean convex, star-shaped hypersurfaces which relates the volume enclosed by a hypersurface to a weighted total mean curvature of the hypers…

The paper studies Finsler manifolds with a new curvature concept.

Simple matrix formulas for Grassmannian curvatures.

Derives new monotone quantities for p-harmonic functions on asymptotically flat 3-manifolds.

New rigidity results for scalar curvature with stabilized conditions.

The study extends conserved quantities theory to non-compact boundary initial data sets.

We prove transverse Weitzenböck identities for the horizontal Laplacians of a totally geodesic foliation. As a consequence, we obtain nullity theorems for the de Rham cohomology assuming only the positivity of curvature quantities transverse to the leaves. Those curvature quantities appear in the adiabatic limit of the…

Proves flows of two-convex Lagrangians are regular, global, and converge.

New curvature measures for 4D manifolds with corners defined and related to Gauss-Bonnet.

Consider a family of smooth immersions $F(\cdot,t): M^n\to \mathbb{R}^{n+1}$ of closed hypersurfaces in $\mathbb{R}^{n+1}$ moving by the mean curvature flow $\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot ν(p,t)$, for $t\in [0,T)$. We show that at the first singular time of the mean curvature flow, certain subcritic…

We define a hierarchy of special classes of constrained Willmore surfaces by means of the existence of a polynomial conserved quantity of some type, filtered by an integer. Type 1 with parallel top term characterises parallel mean curvature surfaces and, in codimension 1, type 1 characterises constant mean curvature su…

Study proves curvature estimates for Kerr spacetime's linearized perturbations.

Study on harmonic functions on nonnegative curvature 3D manifolds.

We introduced an asymptotic quantity that counts area-minimizing surfaces in negatively curved closed 3-manifolds and show that quantity to only be minimized, among all metrics of sectional curvature less than or equal -1, by the hyperbolic metric.

In this paper, we study the line bundle mean curvature flow defined by Jacob and Yau. The line bundle mean curvature flow is a kind of parabolic flows to obtain deformed Hermitian Yang-Mills metrics on a given Kähler manifold. The goal of this paper is to give an $\varepsilon$-regularity theorem for the line bundle mea…

Mass in relativity linked to polyhedra geometry.

The paper extends inequalities to closed Riemannian manifolds.

The paper establishes inequalities for $p$-capacitary functions in flat half-spaces.

Differential quantities, including normals, curvatures, principal directions, and associated matrices, play a fundamental role in geometric processing and physics-based modeling. Computing these differential quantities consistently on surface meshes is important and challenging, and some existing methods often produce …

The paper derives inequalities for $p$-capacitary functions in 3-manifolds with nonnegative scalar curvature.

Global geometric expressions derived for manifold embeddings.

New biharmonic Steklov problem on forms yields eigenvalue estimates.

We revisit the problem of uniqueness for the Ricci flow and give a short, direct proof, based on the consideration of a simple energy quantity, of Hamilton/Chen-Zhu's theorem on the uniqueness of complete solutions of uniformly bounded curvature. With a variation of this quantity and technique, we further prove a uniqu…

Unified view of monotonicity formulas for inverse mean curvature flow and $p$-capacitary potentials.

We prove boundedness and polynomial decay statements for solutions to the spin $\pm2$ generalized Teukolsky system on a Reissner-Nordström background with small charge. The first equation of the system is the generalization of the standard Teukolsky equation in Schwarzschild for the extreme component of the curvature $…

In this paper, we derive the evolution equation for the first eigenvalue of the Witten-Laplace operator acting on the space of functions along the mean curvature flow on a closed oriented manifold. We show some interesting monotonic quantities under the mean curvature flow.

This is the first of two papers where we address and partially confirm a conjecture of Deser and Schwimmer, originally postulated in high energy physics. The objects of study are scalar Riemannian quantities constructed out of the curvature and its covariant derivatives, whose integrals over compact manifolds are invar…

The Gauss-Bonnet inequality holds for certain non-aspherical manifolds up to dimension five.

A geometric interpretation of curvature and torsion of linear transports along paths is presented. A number of (Bianchi type) identities satisfied by these quantities are derived. The obtained results contain as special cases the corresponding classical ones concerning curvature and torsion of linear connections.

This paper continues the study of Alexandrov-Fenchel inequalities for quermassintegrals for $k$-convex domains. It focuses on the application to the Michael-Simon type inequalities for $k$-curvature operators. The proof uses optimal transport maps as a tool to relate curvature quantities defined on the boundary of a do…

We state and prove a Chern-Osserman-type inequality in terms of the volume growth for complete surfaces with controlled mean curvature properly immersed in a Cartan-Hadamard manifold $N$ with sectional curvatures bounded from above by a negative quantity $K_{N}\leq b<0$

We characterize symmetric spaces of non-positive curvature by the equality case of general inequalities between geometric quantities

The paper studies surfaces with spherical curvature lines and their generation by constrained elastic curves.

Computes a new metric quantity Y(M) for Riemannian 2d-manifolds.

We state and prove a Chern-Osserman Inequality in terms of the volume growth for minimal surfaces properly immersed in a Cartan-Hadamard manifold N with sectional curvatures bounded from above by a negative quantity.

Sharp estimates for p-capacity on manifolds with Ricci curvature bounds.

We introduce some new curvature quantities such as conformal Ricci curvature and bi-Ricci curvature and extend the classical Myers theorem under these new curvature conditions. Moreover, we are able to obtain the Myers type theorem for minimal submanifolds in ambient manifolds with positive bi-Ricci curvature. Some top…