arXiv research

Local smoothing of metrics with small curvature, removing Ricci curvature condition.

Study on volume of tubes and concentration in Riemannian geometry.

In this paper we study the local regularity of closed surfaces immersed in a Riemannian 3-manifold flowing by Willmore flow. We establish a pair of concentration-compactness alternatives for the flow, giving a lower bound on the maximal time of existence of the flow proportional to the concentration of the curvature an…

We investigate different concentration-compactness phenomena related to the Q-curvature in arbitrary even dimension. We first treat the case of an open domain in $R^{2m}$, then that of a closed manifold and, finally, the particular case of the sphere $S^{2m}$. In all cases we allow the sign of the Q-curvature to vary, …

This paper presents compact notations for concentration inequalities and convenient results to streamline probabilistic analysis. The new expressions describe the typical sizes and tails of random variables, allowing for simple operations without heavy use of inessential constants. They bridge classical asymptotic nota…

Sharp bounds on quasimode norms on compact space forms.

Study of integral flows on Riemannian manifolds with focus on blow-up profiles and concentration-compactness.

Paper studies equatorial concentration of measure in sphere immersions and submersions.

New operators help focus on specific areas in complex math problems.

Flat space for manifolds with tiny curvature.

The paper shows how solutions of perturbed Dirac operators concentrate near singular sets.

Solves Yamabe problem on compact manifolds using variational methods.

Proves continuity and singular set dimension for 2D maps with Q values.

Study convergence of Yamabe flow on singular spaces with positive constant.

In this work, we prove the existence of a family of solutions of the Allen-Cahn equation with nonlinear Neumann boundary condition under some constraints, whose nodal sets concentrate asymptotically to a given volume nondegenerate capillary hypersurface in a compact Riemannian manifold. Our construction is inspired by …

In this article we examine the concentration and oscillation effects developed by high-frequency eigenfunctions of the Laplace operator in a compact Riemannian manifold. More precisely, we are interested in the structure of the possible invariant semiclassical measures obtained as limits of Wigner measures correspondin…

We study the non-asymptotic behavior of a Coulomb gas on a compact Riemannian manifold. This gas is a symmetric n-particle Gibbs measure associated to the two-body interaction energy given by the Green function. We encode such a particle system by using an empirical measure. Our main result is a concentration inequalit…

Proposes methods for local clustering in attributed graphs.

Mean exit times concentrate near equators and minimal hypersurfaces in high dimensions.

In this paper, I shall demonstrate that sufficiently high-dimensional closed positively-curved Riemannian manifolds are either diffeomorphic to a spherical space form, or isometric to a locally compact rank one symmetric space. This surprising classification of positively-curved Riemannian manifolds results from combin…

We prove several finiteness theorems for the normal bundles to souls in nonnegatively curved manifolds. More generally, we obtain finiteness results for open Riemannian manifolds whose topology is concentrated on compact domains of ``bounded geometry''.

Estimates for eigenfunctions and quasimodes on compact manifolds.

The paper examines stability of Sobolev inequalities on manifolds with Ricci curvature bounds.

The paper studies Dirac operators and their solutions concentrating near singular sets.

Let F be a Kähler foliation on a compact Riemannian manifold M. we study the properties of infinitesimal automorphisms on (M,F), and in particular we concentrate on the transversal conformal field, transversal projective field and transversally holomorphic field

We study perturbed Dirac operators of the form $D_s= D + s{\cal A} :Γ(E)\rightarrow Γ(F)$ over a compact Riemannian manifold $(X, g)$ with symbol $c$ and special bundle maps ${\cal A} : E\rightarrow F$ for $s>>0$. Under a simple algebraic criterion on the pair $(c, {\cal A})$, solutions of $D_sψ=0$ concentrate as $s\t…

Study deformations of compact Calabi-Yau conifolds with singularities.

We exhibit a concentration-collapse decomposition of singularities of fourth order curvature flows, including the $L^2$ curvature flow and Calabi flow, in dimensions $n \leq 4$. The proof requires the development of several new a priori estimates. First, we develop a smoothing result for initial metrics with small ener…

For convex co-compact hyperbolic manifolds $Γ\backslash \mathbb{H}^{n+1}$ for which the dimension of the limit set satisfies $δ_Γ< n/2$, we show that the high-frequency Eisenstein series associated to a point $ξ$ "at infinity" concentrate microlocally on a measure supported by (the closure of) the set of points in the …

In this article, we prove energy quantization for approximate (intrinsic and extrinsic) biharmonic maps into spheres where the approximate map is in $L \log L$. Moreover, we demonstrate that if the $L\log L$ norm of the approximate maps does not concentrate, the image of the bubbles are connected without necks.

The paper constructs local solutions concentrating near singular points of spinors.

We study rays and co-rays in the Wasserstein space $P_p(\mathcal{X})$ ($p > 1$) whose ambient space $\mathcal{X}$ is a complete, separable, non-compact, locally compact length space. We show that rays in the Wasserstein space can be represented as probability measures concentrated on the set of rays in the ambient spac…

We prove an existence theorem for Spin(7)-instantons, which are highly concentrated near a Cayley submanifold; thus giving a partial converse to Tian's foundational compactness theorem. As an application, we show how to construct Spin(7)-instantons on Spin(7)-manifolds with suitable local K3 Cayley fibrations. This rec…

We investigate self-similar solutions to the inverse mean curvature flow in Euclidean space. In the case of one dimensional planar solitons, we explicitly classify all homothetic solitons and translators. Generalizing Andrews' theorem that circles are the only compact homothetic planar solitons, we apply the Hsiung-Min…

Bounds on Steklov eigenvalues for manifolds with boundary.

The title is self-explanatory. We aim to give an easy to read and self-contained introduction to the field of harmonic manifolds. Only basic knowledge of Riemannian geometry is required. After we gave the definition of harmonicity and derived some properties, we concentrate on Z. I. Szabó's proof of Lichnerowicz's conj…

We establish a new estimate for the Ginzburg-Landau energies $E_ε(u)=\int_M\frac{1}{2}|du|^2+\frac{1}{4ε^2}(1-|u|^2)^2$ of complex-valued maps $u$ on a compact, oriented manifold $M$ with $b_1(M)\neq 0$, obtained by decomposing the harmonic component $h_u$ of the one-form $ju:=u^1du^2-u^2du^1$ into an integral and frac…

We show that, the solutions of the isoperimetric problem for small volumes are $C^{2,α}$-close to small spheres. On the way, we define a class of submanifolds called pseudo balls, defined by an equation weaker than constancy of mean curvature. We show that in a neighborhood of each point of a compact riemannian manifol…

Study uniform convergence of random walk Laplacians to diffusion Laplacian on smooth manifolds.

In this paper, we consider the generalized lambda constant and the existence of ground states of the generalized Perelman's W-functional from a variational formulation. One result is concerned with the estimation of the generalized $λ$ constant. The other results are about the existence of ground states of generalized …

We study in this work the existence of minimizing solutions to the critical-power type equation $\triangle_{\textbf{g}}u+h.u=f .u^{\frac{n+2}{n-2}}$ on a compact riemannian manifold in the limit case normally not solved by variational methods. For this purpose, we use a concept of "critical function" that was original…

In this paper, we observe a set of functionals of metrics which are all decrease under the Calabi flow and have uniform lower bound along the flow, which give rise to a set of integral estimates on the curvature flow. Using these estimates, together with weak compactness we obtained in previous papers [8] and [10], we …

We study in this work the existence of minimizing solutions to the critical-power type equation $\triangle_{\textbf{g}}u+h.u = f.u^{\frac{n+2}{n-2}}$ on a compact riemannian manifold in the limit case normally not solved by variational methods. For this purpose, we use a concept of "critical function" that was original…

We prove a general inequality for mixed Hessian measures by global arguments. Our method also yields a simplification for the case of complex Monge-Ampère equation. Exploiting this and using Kołodziej's mass concentration technique we also prove the uniqueness of the solutions to the complex Hessian equation on compact…

The consistency of Fréchet medians is proved for probability measures in proper metric spaces. In the context of Riemannian manifolds, assuming that the probability measure has more than a half mass lying in a convex ball and verifies some concentration conditions, the positions of its Fréchet medians are estimated. It…

Theory for algebraic data on categories via concentration structures.

This is a survey of the theory of complex projective (CP^1) structures on compact surfaces. After some preliminary discussion and definitions, we concentrate on three main topics: (1) Using the Schwarzian derivative to parameterize the moduli space (2) Thurston's parameterization of the moduli space using grafting (3) …

Paper addresses concentration of distances for fractional quasi p-norms, identifying conditions for concentration and anti-concentration.