Related papers: Optimal Integral Pinching Results

Any oriented $4$-dimensional Einstein metric with semi-definite sectional curvature satisfies the pointwise inequality \[ \frac{|s|}{\sqrt{6}}\geq|W^+|+|W^-|, \] where $s$, $W^+$ and $W^-$ are respectively the scalar curvature, the…

We shall prove a new non-vanishing theorem for the stable cohomotopy Seiberg-Witten invariant of connected sums of 4-manifolds with positive first Betti number. The non-vanishing theorem enables us to find many new examples of 4-manifolds…

We investigate the structure of a Finsler manifold of nonnegative weighted Ricci curvature including a straight line, and extend the classical Cheeger-Gromoll-Lichnerowicz splitting theorem. Such a space admits a diffeomorphic,…

We bound from above the expected total Betti number of a high degree random real hypersurface in a smooth real projective manifold. This upper bound is deduced from the equirepartition of critical points of a real Lefschetz pencil…

Some results on existence of global Chebyshev coordinates on a Riemannian manifold or, more generally, on Aleksandrov surface are proved. For instance, if the positive and the negative parts of integral curvature of a Riemannian manifold M…

For complete Riemannian manifolds with vanishing Bach tensor and positive constant scalar curvature, we provide a rigidity theorem characterized by some pointwise inequalities. Furthermore, we prove some rigidity results under an inequality…

This paper contains some vanishing theorems for $L^2$ harmonic forms on complete Riemannian manifolds with a weighted Poincar\'e inequality and a certain lower bound of the curvature. The results are in the spirit of Li-Wang and Lam, but…

In this paper, we present extensions of the classical Bonnet-Myers theorem for Riemannian manifolds with nonnegative Ricci curvature. Our results provide criteria for compactness and a method for estimating the diameter of such manifolds…

We prove the general sharp mean value inequality for non-negative superharmonic functions and its corresponding rigidity, which removes the radius restriction of Schoen-Yau's classical result about this inequality. And we obtain an explicit…

Hamilton's pinching conjecture, that three-dimensional complete non-compact manifolds with pinched Ricci curvature are flat, has recently been resolved using Ricci flow. In this paper we prove a direct analogue of that result in all…

In this paper, we give a generalization of the Chern-Lashof theorem for submanifolds with singularities called frontals in Euclidean space. We prove that, for an $n$-dimensional admissible compact frontal in $(n+r)$-dimensional Euclidean…

We use the Gauss-Bonnet theorem and the triangle comparison theorems of Rauch and Toponogov to show that on compact Riemann surfaces of negative curvature period integrals of eigenfunctions $e_\lambda$ over geodesics go to zero at the rate…

A classical theorem of Bochner asserts that the isometry group of a compact Riemannian manifold with negative Ricci curvature is finite. In this paper we give several extensions of Bochner's theorem by allowing "small" positive Ricci…

We propose a notion of integral Menger curvature for compact, $m$-dimensional sets in $n$-dimensional Euclidean space and prove that finiteness of this quantity implies that the set is $C^{1,\alpha}$ embedded manifold with the H{\"o}lder…

We will provide bounds on both the Betti numbers and the torsion part of the homology of hyperbolic orbifolds. These bounds are linear in the volume and are a direct consequence of an efficient simplicial model of the thick part, which we…

In this article, we introduce a $2$-parameter family of affine connections and derive the Ricci curvature. We first establish an integral Bochner technique. On one hand, this technique yields a new proof to our recent work in \cite{LX} for…

We formulate and prove Cutkosky's Theorem regarding the discontinuity of Feynman integrals in the massive one-loop case up to the involved intersection index. This is done by applying the techniques to treat singular integrals developed in…

We establish a new algebraic characterization of sectional curvature bounds $\sec\geq k$ and $\sec\leq k$ using only curvature terms in the Weitzenb\"ock formulae for symmetric $p$-tensors. By introducing a symmetric analogue of the…

In this paper, we prove a quantitative relative index theorem. It provides a conceptual framework for studying some conjectures and open questions of Gromov on positive scalar curvature. More precisely, we prove a $\lambda$-Lipschitz…

In this paper, we proved the Normal Scalar Curvature Conjecture and the Bottcher-Wenzel Conjecture. We also established some new pinching theorems for minimal submanifolds in spheres.