arXiv research

The study examines how average scalar curvature influences geometric properties of Riemannian manifolds.

New bounds for average graph distance using curvature and centrality.

The study finds the minimum average area ratio on hyperbolic manifolds and its relation to scalar curvature.

Estimates scalar curvature on moduli space as genus grows.

Derives curvature formulas for convex metric sums and conditions for positive average variation.

Minimal surfaces and average area ratio found to be maximized by hyperbolic metrics.

Lower bounds on average normal curvature for submanifolds in Riemannian domains.

We prove that S^2 x S^2 satisfies an intermediate condition between having metrics with positive Ricci and positive sectional curvature. Namely, there exist metrics for which the average of the sectional curvatures of any two planes tangent at the same point, but separated by a minimum distance in the 2-Grassmannian, i…

New Ricci curvature means derived from plane curvatures.

Let $(M, g)$ be a complete non-compact Kähler manifold with non-negative and bounded holomorphic bisectional curvature. Extending our techniques developed in \cite{CT3}, we prove that the universal cover $\wt M$ of $M$ is biholomorphic to $\ce^n$ provided either that $(M, g)$ has average quadratic curvature decay, or $…

We compute curvatures of a three-manifold formed by a Weil-Petersson geodesic in Teichmuller space.

In this paper we obtain three results concerning the geometry of complete noncompact positively curved Kähler manifolds at infinity. The first one states that the order of volume growth of a complete noncompact Kähler manifold with positive bisectional curvature is at least half of the real dimension (i.e., the complex…

Sharp gradient estimate for scalar curvature on 3-manifolds.

We write the Euler characteristic X(G) of a four dimensional finite simple geometric graph G=(V,E) in terms of the Euler characteristic X(G(w)) of two-dimensional geometric subgraphs G(w). The Euler curvature K(x) of a four dimensional graph satisfying the Gauss-Bonnet relation sum_x K(x) = X(G) can so be rewritten as …

In this paper, by combining techniques from Ricci flow and algebraic geometry, we prove the following generalization of the classical uniformization theorem of Riemann surfaces. Given a complete noncompact complex two dimensional Kähler manifold $M$ of positive and bounded holomorphic bisectional curvature, suppose its…

The abstract discusses metrics with positive biorthogonal curvature on 5-manifolds.

The paper extends a prediction method to curved spaces.

Proof of Graustein's theorem in different geometries.

Average signature measures geodesics in Lie groups.

We classify, up to homeomorphisms, the closed simply-connected 4-manifolds that admit a Riemannian metric for which averages of pairs of sectional curvatures of orthogonal planes are positive.

New stability analysis improves generalization of multipass SGD.

The study uses Ricci flow to prove flatness of certain Riemannian manifolds.

Simplified Ricci curvature for spherical fluid dynamics models.

Upper bounds on Betti numbers via curvature constraints.

We discuss some consequences of the existence of the holomorphic quadratic Hopf differential on a conformally immersed constant mean curvature topological disc with analytic boundary. In particular, we derive a formula for the mean curvature as a weighted average of the normal curvature of the boundary curve, and a con…

Given a Finsler space (M,F) on a manifold M, the averaging method associates to Finslerian geometric objects affine geometric objects} living on $M$. In particular, a Riemannian metric is associated to the fundamental tensor $g$ and an affine, torsion free connection is associated to the Chern-Rund connection. As an il…

In this paper, we prove that on every Finsler $n$-sphere $(S^n, F)$ for $n\ge 6$ with reversibility $λ$ and flag curvature $K$ satisfying $(\fracλ{λ+1})^2<K\le 1$, either there exist infinitely many prime closed geodesics or there exist $[\frac{n}{2}]-2$ closed geodesics possessing irrational average indices. If in add…

Magnetic Brunn-Minkowski inequalities on Riemannian manifolds

We study the asymptotic behavior of the Kähler-Ricci flow on Kähler manifolds of nonnegative holomorphic bisectional curvature. Using these results we prove that a complete noncompact Kähler manifold with nonnegative bounded holomorphic bisectional curvature and maximal volume growth is biholomorphic to complex Euclide…

Study on compact Kähler surfaces for sign-changing curvatures.

Discretizations of the mean curvature and extrinsic curvature components are constructed on piecewise flat simplicial manifolds, giving approximations for smooth curvature values in a mostly mesh-independent way. These constructions are given in combinatoric form in terms of the extrinsic hinge angles, the intrinsic st…

TD(0) with Polyak-Ruppert averaging achieves robust and fast convergence rates

The paper improves inequalities for Kähler-Einstein manifolds using curvature conditions.

Anosov geodesic flow proven in non-compact manifolds with negative curvature.

In topological data analysis, persistent homology is used to study the "shape of data". Persistent homology computations are completely characterized by a set of intervals called a bar code. It is often said that the long intervals represent the "topological signal" and the short intervals represent "noise". We give ev…

Let $(M^n, g)$ be a complete non-compact Kähler manifold with non-negative and bounded holomorphic bisectional curvature. We prove that $M$ is holomorphically covered by a pseudoconvex domain in $\C^n$ which is homeomorphic to $\R^{2n}$, provided $(M^n, g)$ has uniform linear average quadratic curvature decay.

Study on 3-manifolds with nonnegative scalar curvature and positive harmonic functions.

Graphs with bounded degrees and non-negative Ollivier-Ricci curvature have subexponential growth and diffusive random walk.

New theorem limits curvature of Einstein manifolds.

We study a $(k+1)$-dimensional hyperbolic space of a negative constant sectional curvature $κ=-1/ρ^2$. Let $λ$ be a real eigenvalue and $f_λ (x)$ be an eigenfunction of the hyperbolic Laplacian assuming a non-zero value at $x_0$. Then the average value of $f_λ(x)$ over any sphere centered at $x_0$ allows to identify th…

We prove a vanishing and estimation theorem for the $p^{\text{th}}$-Betti number of closed $n$-dimensional Riemannian manifolds with a lower bound on the average of the lowest $n-p$ eigenvalues of the curvature operator. This generalizes results due to D. Meyer, Gallot-Meyer, and Gallot. For example, in dimensions $n=5…

Kodaira embedding theorem provides an effective characterization of projectivity of a Kähler manifold in terms the second cohomology. Recently X. Yang [21] proved that any compact Kähler manifold with positive holomorphic sectional curvature must be projective. This gives a metric criterion of the projectivity in terms…

Study shows how curved surfaces evolve smoothly to spherical shapes.

Sophie Germain's mean curvature deserves recognition as a surface shape measure.

Using the method of De Lellis-Topping, we prove some almost Schur type results. For example, one of our results gives a quantitative measure of how close the higher mean curvature of a submanifold is to its average value. We also derive another sharp Andrews-De Lellis-Topping type inequality involving the Riemannian cu…

The paper rethinks the use of exponential averaging in machine learning optimization.

Enhanced Bishop-Gromov theorem for homogeneous and inhomogeneous spaces.

We shall prove that under some volume growth condition, the essential spectrum of the Laplacian contains the interval $[(n-1)^2K/4, \infty)$ if an $n$-dimensional Riemannian manifold has an end and the average of the part of the Ricci curvature on the end which lies below a nonpositive constant $(n-1)K$ converges to ze…