arXiv research

Study on RCD(0,N) spaces with small linear diameter growth.

Since non-compact RCD(0, N) spaces have at least linear volume growth, we study noncompact RCD(0, N) spaces with linear volume growth in this paper. One of the main results is that the diameter of level sets of a Busemann function grow at most linearly on a noncompact RCD(0, N) space satisfying the linear volume growth…

Study on topological properties and boundaries of RCD(K,N) spaces.

Study spacelike loxodromes on helicoidal surfaces in Lorentzian n-space.

The paper generalizes rectifying and normal curves in Lorentzian n-space.

The study presents examples of $CD(0,N)$ spaces with varying dimensions and discusses the limitations of the $CD(0,N)$ condition.

We show that the constant mean curvature hypersurfaces in the hyperbolic n-space spanning the boundary of a star shaped C^{1,1} domain in the asymptotic sphere give a foliation of the hyperbolic n-space. We also show that if C is a closed codimension-1 C^{2,a} submanifold in the asymptotic sphere bounding a unique cons…

In this paper, we investigate the similarity transformations in the Minkowski-n space. We study the geometric invariants of non-null curves under the similarity transformations. Besides, we extend the fundamental theorem for a non-null curve according to a similarity motion. We determine all non-null self-similar curve…

Study heat content on RCD(K,N) spaces with specific boundary conditions.

Extends Margulis Lemma to RCD(K,N) spaces.

Abstract shows entropy and convexity definitions of very strict $CD(K,N)$ spaces are equivalent.

In this paper, we prove pointwise convergence of heat kernels for mGH-convergent sequences of $RCD^*(K,N)$-spaces. We obtain as a corollary results on the short-time behavior of the heat kernel in $RCD^*(K,N)$-spaces. We use then these results to initiate the study of Weyl's law in the $RCD$ setting

This study proves the Half Space Property for RCD(0,N) and RCD(K,N) spaces.

Sharp log-Sobolev inequalities proved for ${\sf CD}(0,N)$ spaces.

The main results of this paper consists of two parts. Firstly, we obtain an almost rigidity theorem which says that on a RCD(0, N) space, when a domain between two level sets of a distance function has almost maximal volume compared to that of a cylinder, then this portion is close to a cylinder as a metric space. Seco…

Sharp Talenti-type comparison theorem for p-Laplacian on RCD(K,N) spaces.

Compact RCD(K,N) spaces with maximal rank are homeomorphic to infranilmanifolds.

Study characterizes non-collapsed RCD(K, N) spaces using heat kernel metrics.

Proves metric measure spaces with certain properties are one-dimensional.

The main goal of the paper is to prove the existence of the universal cover for $RCD^*(K,N)$-spaces. This generalizes earlier work of C. Sormani and the second named author on the existence of universal covers for Ricci limit spaces. As a result, we also obtain several structure results on the (revised) fundamental gro…

In the asymmetric setting, Hilbert's fourth problem asks to construct and study all (non-reversible) projective Finsler metrics: Finsler metrics defined on open, convex subsets of real projective $n$-space for which geodesics lie on projective lines. While asymmetric norms and Funk metrics provide many examples of esse…

Embeds RCD*(K,N) spaces into L^2 via eigenmaps.

The paper studies properties of $\mathrm{RCD}(K,N)$ spaces and their boundaries.

Paper extends Weyl's lemma to RCD(K,N) spaces.

Study shows how maps from certain geometric spaces behave near their edges.

We prove that $n$-dimensional ($n\geqslant3$) complete and non-compact metric measure spaces with non-negative weighted Ricci curvature in which some Caffarelli-Kohn-Nirenberg type inequality holds are close to the model metric measure $n$-space (i.e., the Euclidean metric $n$-space).

Entropy derived from Colding's volume on Ricci-flat manifolds.

Compact RCD spaces are proven to be smooth manifolds under specific harmonicity conditions.

Given a compact Alexadrov $n$-space $Z$ with curvature curv $\ge κ$, and let $f: Z\to X$ be a distance non-increasing onto map to another Alexandrov $n$-space with curv $\ge κ$. The relative volume rigidity conjecture says that if $X$ achieves the relative maximal volume i.e. $vol(Z)=vol(X)$, then $X$ is isometric to $…

Study fundamental groups of RCD spaces without smoothness or curvature bounds.

Sharp eigenvalue bounds on metric measure spaces extend Cheng's theorem.

Heat kernels map RCD spaces to Riemannian manifolds.

Study rectifiability of finite perimeter sets in RCD(K,N) spaces.

We show that if a noncollapsed $CD(K,n)$ space $X$ with $n\ge 2$ has curvature bounded above by $κ$ in the sense of Alexandrov then $K\le (n-1)κ$ and $X$ is an Alexandrov space of curvature bounded below by $K-κ(n-2)$. We also show that if a $CD(K,n)$ space $Y$ with finite $n$ has curvature bounded above then it is inf…

The paper studies the limits and rigidity of almost homogeneous spaces with Ricci curvature bounds.

In this paper, in Euclidean n -space, we investigate the relation between slant helices and spherical helices. Moreover, in E n, we show that a slant helix and the tangent indicatrix of the slant helix have the same axis (or direction). Also, we give the important relations between slant helices, spherical helices in E…

New isoperimetric inequality for clamped plates in RCD(0,N) spaces, sharp and stable.

Researchers extend monotonicity formulas for harmonic functions in RCD(0,N) spaces.

We prove that if the dimension of the first cohomology group of a $RCD^*(0,N)$ space is $N$, then the space is a flat torus. This generalizes a classical result due to Bochner to the non-smooth setting and also provides a first example where the study of the cohomology groups in such synthetic framework leads to geomet…

Sharp gradient estimate for Green functions on non-parabolic RCD(0,N) spaces.

We show that in any infinitesimally Hilbertian $CD^*(K,N)$-space at almost every point there exists a Euclidean weak tangent, i.e. there exists a sequence of dilations of the space that converges to a Euclidean space in the pointed measured Gromov-Hausdorff topology. The proof follows by considering iterated tangents a…

Let $(M,g)$ be a smooth Riemannian manifold and $\mathsf{G}$ a compact Lie group acting on $M$ effectively and by isometries. It is well known that a lower bound of the sectional curvature of $(M,g)$ is again a bound for the curvature of the quotient space, which is an Alexandrov space of curvature bounded below. Moreo…

We prove that for every analytic curve in the complex plane, Euclidean and spherical arc-lengths are global conformal parameters. We also prove that for any analytic curve in the hyperbolic plane, hyperbolic arc-length is also a global parameter. We generalize some of these results to the case of analytic curves in Euc…

The paper studies the geometry of multi-particle Schrödinger semigroups using $\mathrm{RCD}^*(K,N)$ spaces.

Paper proves Hölder continuity of tangent cones in RCD(K,N) spaces.

We are interested in contractible n-manifolds M which "split" as M = A union B where A,B, and A intersect B are all homeomorphic to Euclidean n-space (such M are called open n-splitters) or A,B, and A intersect B are all homeomorphic to the n-dimensional unit ball (such M are called closed n-splitters). We introduce a …

Survey explains Kahler-Einstein metrics construction from interpolation problems.

In this paper, we partially settle down the long standing open problem of the finite time blow-up property about the nonlinear Schr$\ddot{o}$dinger equations on some Riemannian manifolds like the standard 2-sphere $S^2$ and the hyperbolic 2-space $H^{2}(-1)$. Using the similar idea, we establish such blow-up results on…