arXiv research

The paper proves new inequalities on the unit ball in higher dimensions.

Sharp characterization of Willmore invariant in higher dimensions.

We introduce and study the conical curvature-dimension condition, $CCD(K,N)$, for graphs. We show that $CCD(K,N)$ provides necessary and sufficient conditions for the underlying graph to satisfy a sharp global Poincaré inequality which in turn translates to a sharp lower bound for the first eigenvalues of these graphs.…

Sharp stability in Almgren problem solved in any dimension.

New estimates show spectral gap stability in RCD spaces, close to Beta distribution.

Sharp regularity for Pfaff system leads to isometric immersions in arbitrary dimensions.

Sharp bound on singular set dimension for specific geometric problems.

New method for Sharpe ratio analysis in high dimensions using residual-based nodewise regression.

Sharp bounds on Alexandrov spaces' boundaries with rigidity analysis.

Sharp estimate for flow in any dimension.

We obtain new sharp isoperimetric inequalities on a Riemannian manifold equipped with a probability measure, whose generalized Ricci curvature is bounded from below (possibly negatively), and generalized dimension and diameter of the convex support are bounded from above (possibly infinitely). Our inequalities are shar…

Sharp inequality in spaces with non-negative Ricci curvature.

The paper studies how submanifolds of a sphere evolve over time.

In this paper, we give a sharp lower bound for the first (nonzero) Neumann eigenvalue of Finsler-Laplacian in Finsler manifolds in terms of diameter, dimension, weighted Ricci curvature.

Sharp dimension constraints for positive intermediate curvature metrics are established.

Generalized Blaschke rolling theorem for curved spaces.

Formula calculates instanton homology dimensions for knot surgeries over arbitrary fields.

In this paper, we extend the sharp lower bounds of spectal gap, due to Chen- Wang [10, 11], Bakry-Qian [6] and Andrews-Clutterbuck [5], from smooth Riemaniannian manifolds to general metric measure spaces with Riemannian curvature-dimension condition RCD*(K;N).

Sharp spectral theorem splits certain non-compact manifolds.

Study on polynomial growth functions and forms on gradient Ricci solitons.

This is the very first paper to focus on the CR analogue of Yau's uniformization conjecture in a complete noncompact pseudohermitian $(2n+1)$-manifold of vanishing torsion (i.e. Sasakian manifold) which is an odd dimensional counterpart of Kähler geometry. In this paper, we mainly deal with the problem of the sharp dim…

Proves a theorem for Assouad dimension with applications to distance sets and radial projections.

Sharp gradient estimates for positive Ricci curvature manifolds.

For a sequence of blow up solutions of the Yamabe equation on non-locally confonformally flat compact Riemannian manifolds of dimension 10 or 11, we establish sharp estimates on its asymptotic profile near blow up points as well as sharp decay estimates of the Weyl tensor and its covariant derivatives at blow up points…

Sharp area estimates for minimal submanifolds in curved spaces.

Interprets LSTM solar flare predictions using SHARP parameters.

Formula established for instanton homology of dual knots.

We prove a Sobolev inequality which holds on submanifolds in Euclidean space of arbitrary dimension and codimension. This inequality is sharp if the codimension is at most 2. As a special case, we obtain a sharp isoperimetric inequality for minimal submanifolds in Euclidean space of codimension at most 2.

Let $M^n$ be a complete noncompact K$\ddot{a}$hler manifold of complex dimension $n$ with nonnegative holomorphic bisectional curvature. Denote by $\mathcal{O}$_d(M^n)$the space of holomorphic functions of polynomial growth of degree at most$d$on$M^n$. In this paper we prove that$dim_{\mathbb{C}}{\mathcal{O}}_d(…

The paper examines stability of the Sobolev inequality in metric spaces with curvature dimension conditions.

Sharp stability of Möbius group among sphere-valued maps proved in arbitrary dimensions.

We establish in this paper an upper bound on the second eigenvalue of n-dimensional spheres in the conformal class of the round sphere. This upper bound holds in all dimensions and is asymptotically sharp as the dimension increases.

We obtain some improved essentially sharp Kakeya-Nikodym estimates for eigenfunctions in two-dimensions. We obtain these by proving stronger related microlocal estimates involving a natural decomposition of phase space that is adapted to the geodesic flow.

Sharp curvature estimates for expanding Ricci solitons in various dimensions.

Sharp bounds for eigenfunctions on product spaces restricted to submanifolds.

The paper sets lower bounds for Laplacian eigenvalues and clamped plate problem eigenvalues.

Given a closed Riemannian manifold $(N^{n+1},g)$, $n+1 \geq 3$ we prove the compactness of the space of singular, minimal hypersurfaces in $N$ whose volumes are uniformly bounded from above and the $p$-th Jacobi eigenvalue $λ_p$'s are uniformly bounded from below. This generalizes the results of Sharp and Ambrozio-Carl…

Sharp Lipschitz bounds for flow-matching and diffusion models with optimal sampling rates.

We classify local minimizers of $\intσ_2+\oint H_2$ among all conformally flat metrics in the Euclidean $(n+1)$-ball, $4\leq n\leq 5$, for which the boundary has unit volume, subject to an ellipticity assumption. We also classify local minimizers of the analogous functional in the critical dimension $n+1=4$. If minimiz…

Sharp upper bounds found for Steklov eigenvalues of a specific hypersurface.

New theory explains how chaotic training improves neural network generalization.

We show a sharp conformally invariant gap theorem for Yang-Mills connections in dimension 4 by exploiting an associated Yamabe-type problem.

Motivated by applications in model-free finance and quantitative risk management, we consider Fréchet classes of multivariate distribution functions where additional information on the joint distribution is assumed, while uncertainty in the marginals is also possible. We derive optimal transport duality results for the…

Sharp bounds on K-semistable Fano varieties for low dimensions.

Sharp estimates link curvature to topology, proving manifold rigidity.

Perimeter minimizers in curved spaces have a singular set no more than 5 dimensions.

New algorithm optimizes convex functions with noisy evaluations in one dimension.

Tian's criterion for K-stability states that a Fano variety of dimension $n$ whose alpha invariant is greater than $\frac{n}{n+1}$ is K-stable. We show that this criterion is sharp by constructing singular Fano varieties with alpha invariants $\frac{n}{n+1}$ that are not K-polystable for sufficiently large $n$. We also…