arXiv research

Sharp comparison theorems for 3D manifolds with scalar curvature bound.

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

In this paper, by using the Bochner technique on almost Hermitian manifolds, we obtain a complex Hessian comparison for almost Hermitian manifolds generalizing the Laplacian comparison for almost Hermitian manifolds by Tossati, and reprove a diameter estimate for almost Hermitian manifolds by Gray. Moreover, we obtain …

Sharp inequality for compactifying Poincaré-Einstein manifolds.

We prove a comparison theorem on the first Neumann eigenvalue on Bakry-Emery manifolds. Examples are constructed to illustrate the sharpness of the result. A linear explicit lower bound is also proved. We also discuss the asymptotic sharpness of such a result.

Sharp estimates for parabolic equations on manifolds using symmetrization.

Sharp gradient estimates for positive Ricci curvature manifolds.

Study Brownian motions and heat kernel bounds on Kähler and quaternion Kähler manifolds.

We prove a comparison theorem for the compact surfaces with negative Euler characteristic via the Ricci flow.

Sharp comparison for sub-Gaussian random variables in convex order.

The paper develops new methods to study sharp isoperimetric properties on complex spaces.

The paper studies volume and area comparisons in non-compact 3-manifolds with non-negative scalar curvature.

Sharp inequality proved in 3D hyperbolic spaces using flow methods.

Paper proves a sharp weighted Isoperimetric inequality for substatic manifolds.

The paper explores sharp isoperimetric properties on non-compact spaces with Ricci bounds.

Sharp bounds and rigidity theorems for eigenvalues on manifolds.

Study eigenvalues of p-Laplacian on manifolds with Robin boundary conditions.

We develop a variational theory of geodesics for the canonical variation of the metric of a totally geodesic foliation. As a consequence, we obtain comparison theorems for the horizontal and vertical Laplacians. In the case of Sasakian foliations, we show that sharp horizontal and vertical comparison theorems for the s…

Sharp chord-arc estimates for curve shortening flow on spheres.

We define a notion of renormalized volume of an asymptotically hyperbolic manifold. Moreover, we prove a sharp volume comparison theorem for metrics with scalar curvature at least -6. Finally, we show that the inequality is strict unless the metric is isometric to one of the Anti-deSitter-Schwarzschild metrics.

On Kahler manifolds with Ricci curvature lower bound, assuming the real analyticity of the metric, we establish a sharp relative volume comparison theorem for small balls. The model spaces being compared to are complex space forms, i.e, Kahler manifolds with constant holomorphic sectional curvature. Moreover, we give a…

On H-type sub-Riemannian manifolds we establish sub-Hessian and sub-Laplacian comparison theorems which are uniform for a family of approximating Riemannian metrics converging to the sub-Riemannian one. We also prove a sharp sub-Riemannian Bonnet-Myers theorem that extends to this general setting results previously pro…

We prove comparison theorems for the sub-Riemannian distortion coefficients appearing in interpolation inequalities. These results, which are equivalent to a sub-Laplacian comparison theorem for the sub-Riemannian distance, are obtained by introducing a suitable notion of sub-Riemannian Bakry-Émery curvature. The model…

We complete the picture of sharp eigenvalue estimates for the p-Laplacian on a compact manifold by providing sharp estimates on the first nonzero eigenvalue of the nonlinear operator $Δ_p$ when the Ricci curvature is bounded from below by a negative constant. We assume that the boundary of the manifold is convex, and p…

Motivated by a classical comparison result of J. C. F. Sturm we introduce a curvature-dimension condition CD(k,N) for general metric measure spaces and variable lower curvature bound k. In the case of non-zero constant lower curvature our approach coincides with the celebrated condition that was proposed by K.-T. Sturm…

In this paper we prove that given a volume, among all domains with smooth boundary in rank-1 symmetric spaces of noncompact type, geodesic balls maximizes the first nonzero Steklov eigenvalue. We also prove a comparison result for the first nonzero Steklov eigenvalue for domains in simply connected Riemannian manifolds…

Using McCann's transportation map, we establish a transport inequality on compact manifolds with positive Ricci curvature. This inequality contains the sharp spectral comparison estimates.

We obtain sharp quantitative Laplacian upper and lower estimates under no assumption on curvatures. As a result, we derive quantitative Laplacian, area and volume comparison theorems for tubes in Riemannian and Kähler manifolds under weak integral curvature assumptions. We also give some applications, such as a general…

The paper proves properties of Lipschitz spacetimes with bounded Ricci curvature.

Sharp inequality found for hypersurfaces in curved spaces.

The study establishes inequalities on Finsler manifolds with weighted Ricci curvature.

We prove the sharp estimate on the first nonzero eigenvalue of the p-laplacian on a compact Riemannian manifold with nonnegative Ricci curvature and possibly with convex boundary (in this case we assume Neumann b.c. on the p-laplacian). The proof is based on a gradient comparison theorem. We will also charachterize the…

Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.

The paper explores dualities in differential equations and their applications in Riemannian geometry.

Sharp heat kernel estimates on manifolds lead to solutions of the Parabolic Anderson model.

Paper compares total quotient curvature and proves bounds for Einstein metric.

Sharp estimates derived for quasilinear equations on metric measure spaces.

In an incomplete market, including liquidly-traded European options in an investment portfolio could potentially improve the expected terminal utility for a risk-averse investor. However, unlike the Sharpe ratio, which provides a concise measure of the relative investment attractiveness of different underlying risky as…

The paper proves gradient and comparison inequalities for RCD spaces.

Let $(X,d,μ)$ be a $RCD^\ast(K, N)$ space with $K\in \mathbb{R}$ and $N\in [1,\infty]$. For $N\in [1,\infty)$, we derive the upper and lower bounds of the heat kernel on $(X,d,μ)$ by applying the parabolic Harnack inequality and the comparison principle, and then sharp bounds for its gradient, which are also sharp in t…

New comparison theorem for submanifolds with geometric inequalities.

In this paper, based on the local comparison principle in [12], we study the local behavior of the difference of two spacelike graphs in a neighborhood of a second contact point. Then we apply it to the constant mean curvature equation in 3-dimensional Lorentz-Minkowski space $\mathbb{L}^3$ and get the uniqueness of cr…

We give a sharp comparison between the spectra of two Riemannian manifolds (Y,g) and (X,g_0) under the following assumptions: (X,g_0) has bounded geometry, (Y,g) admits a continuous Gromov-Hausdorff ε-approximation onto (X,g_0) of non zero absolute degree, and the volume of (Y,g) is almost smaller than the volume of (X…

We study the biharmonic Steklov eigenvalue problem on a compact Riemannian manifold $Ω$ with smooth boundary. We give a computable, sharp lower bound of the first eigenvalue of this problem, which depends only on the dimension, a lower bound of the Ricci curvature of the domain, a lower bound of the mean curvature of i…

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

Develops comparison methods for semilinear elliptic problems on Riemannian manifolds with Ricci lower bound.

SyncRank recovers global ranking from noisy comparisons with theoretical guarantees.

Sharp Steklov eigenvalue estimates for differential forms on manifolds.