arXiv research

Proves gap rigidity theorem for Hermitian symmetric spaces.

The paper proves gap theorems for Yang-Mills on manifolds with positive Yamabe.

The paper proves gap theorems for Yang-Mills theory on specific manifolds.

The paper proves a gap theorem for special harmonic maps between spheres.

The paper extends gap theorems for Bach-flat 4-manifolds.

Abstract: Unknown status of Jacobian Conjecture, proof has a gap.

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

The study proves a gap theorem for shrinking gradient Ricci solitons with specific curvature and volume conditions.

New theorem shows curvature concentration depends linearly on volume ratio.

The paper proves lower bounds for Gaussian-weighted curvature integrals of self-shrinkers.

In this paper, we prove a classification theorem for self-shrinkers of the mean curvature flow with $|A|^2\le 1$ in arbitrary codimension. In particular, this implies a gap theorem for self-shrinkers in arbitrary codimension.

Local gap theorem for Ricci shrinkers ensures flatness if certain functionals are close to zero.

Local gaps in Ricci shrinkers depend only on dimension.

Lu conjecture proven for minimal 2-spheres and surfaces under certain conditions.

Sharp convergence theorem for Yang-Mills flow on ALE manifolds proved.

This paper extends gap theorems for submanifolds in hyperbolic space.

Study proves rigidity and gap theorems for specific metrics.

In this paper we prove a gap theorem for Kähler manifolds with nonnegative orthogonal bisectional curvature and nonnegative Ricci curvature, which generalizes an earlier result of the first author. We also prove a Liouville theorem for plurisubharmonic functions on such a manifolds, which generalizes a previous result …

Study on automorphisms of K3 and Enriques surfaces, proving entropy gaps and achirality.

Adapting the method of Andrews-Clutterbuck we prove an eigenvalue gap theorem for a class of non symmetric second order linear elliptic operators on a convex domain in euclidean space. The class of operators includes the Bakry-Emery laplacian with potential and any operator with second order term the laplacian whose fi…

Corrects a proof gap in the Morse lemma for hyperbolic spaces.

New curvature measure defined for graphs, with bounds on diameter and spectral gap.

In this paper we develop a bubble tree structure for a degenerating class of Riemannian metrics satisfying some global conformal bounds on compact manifolds of dimension 4. Applying the bubble tree structure, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and a diameter bound f…

Abstract: Study continuous maps between Kähler manifolds, proving a gap theorem.

Gradient steady Ricci solitons are natural generalizations of Ricci-flat manifolds. In this article, we prove a curvature gap theorem for gradient steady Ricci solitons with nonconstant potential functions; and a curvature gap theorem for Ricci-flat manifolds, removing the volume growth assumptions in known results.

In this paper, we study the properties of the first global term in the polyhomogeneous expansions for Liouville's equation. We obtain rigidity and gap results for the boundary integral of the global coefficient. We prove that such a boundary integral is always nonpositive, and is zero if and only if the underlying doma…

In the present paper, by using estimates for the generalized Ricci curvature, we shall give some gap theorems for Ricci-harmonic solitons showing some necessary and sufficient conditions for the solitons to be harmonic-Einstein. Our results may be regarded as a generalization of recent works by H. Li, and M. Fernandez-…

The study proves a gap theorem for CAT(0) spaces with a constant below 1/(6√π).

The paper proves gap results for self-shrinkers in $r$-mean curvature flow.

Proves convexity of certain hypersurfaces with negative λ.

Sharp curvature estimates for expanding Ricci solitons in various dimensions.

The paper sets limits on the number of ends of certain geometric structures.

We develop some estimates under the Ricci flow and use these estimates to study the blowup rates of curvatures at singularities. As applications, we obtain some gap theorems: $\displaystyle \sup_X |Ric|$ and $\displaystyle \sqrt{\sup_X |Rm|} \cdot \sqrt{\sup_X |R|}$ must blowup at least at the rate of type-I. Our estim…

New theorem improves spectral gap for sampling from mixture distributions.

New gap theorem for hypersurfaces with constant mean curvature in space forms.

In this paper, we show that any ancient solution to the Ricci flow with the reduced volume whose asymptotic limit is sufficiently close to that of the Gaussian soliton is isometric to the Euclidean space for all time. This is a generalization of Anderson's result for Ricci-flat manifolds. As a corollary, a gap theorem …

The paper proves parabolic gap theorems for Yang-Mills energy.

Paper studies a new curvature system and proves rigidity and gap theorems.

A new topological gap theorem improves the systole of 3-manifolds with positive scalar curvature.

In this short note, we find a new gap phenomena on Riemannian manifolds, which says that for any complete noncompact Riemannian manifold with nonnegative Ricci curvature, if the scalar curvature decays faster than quadratically, then it is Ricci flat.

The Whitney sphere has a unique energy gap for a specific equation.

The paper proves various inequalities on gradient shrinking Ricci solitons.

Polynomial density theorem for specific subgroup orbits in quotient spaces.

The study proves unique and isolated properties of Einstein 5-manifolds via gap theorems in 4 dimensions.

The classical Three Gap Theorem asserts that for a natural number n and a real number p, there are at most three distinct distances between consecutive elements in the subset of [0,1) consisting of the reductions modulo 1 of the first n multiples of p. Regarding it as a statement about rotations of the circle, we find …

In this note we fill a gap in the proof of the main theorem (Theorem 1.2) of our paper 'Surfaces in 4-manifolds', Math. Res. Letters 4 (1997), 907-914.

We prove a Liouville theorem for the plurisubharmonic functions on complete Kaelher manifolds. As the applications, we prove a splitting theorem for complete Kaehler manifolds with nonnegative biscetional curvature in terms of the linear growth harmonic functions and a optomal gap theorem for such manifolds.

Study on invariant Seifert surfaces for strongly invertible knots, showing large gaps in genus.