arXiv research

Study on stable minimal hypersurfaces under Ricci curvature constraints.

Motivated by the celebrated Schoen-Yau-Gromov-Lawson surgery theory on metrics of positive scalar curvature, we construct a double manifold associated with a minimal isoparametric hypersurface in the unit sphere. The resulting double manifold carries a metric of positive scalar curvature and an isoparametric foliation …

We prove an analog of the Schoen-Yau univalentness theorem for saddle maps between discs.

Study finds open manifolds without complete metrics with positive scalar curvature.

Schoen-Yau's zero mass theorem stability remains an open question.

Extends positive mass theorem to arbitrary dimensions using a new inductive scheme.

Based on the work of Schoen-Yau, we derive an estimate of the first eigenvalue of a Schrödinger Operator (the Jaocbi operator of minimal surfaces in flat 3-spaces) on surfaces.

We give a short proof of the systolic inequality for the n-dimensional torus. The proof uses minimal hypersurfaces. It is based on the Schoen-Yau proof that an n-dimensional torus admits no metric of positive scalar curvature.

Explicit mass bound for 3D asymptotically flat manifolds using harmonic functions.

Proves non-existence of metrics with positive curvature for certain connected sums.

Let $(Y,g)$ be a compact Riemannian manifold of positive scalar curvature (psc). It is well-known, due to Schoen-Yau, that any closed stable minimal hypersurface of $Y$ also admits a psc-metric. We establish an analogous result for stable minimal hypersurfaces with free boundary. Furthermore, we combine this result wit…

Vacuum gravity shows black holes can form without collapse.

We find two conditions related to the {\it news functions} of the Bondi's radiating vacuum spacetimes. We provide a complete proof of the positivity of the Bondi mass by using Schoen-Yau's method under one condition and by using Witten's method under another condition.

In this paper, we prove a classification theorem of 4-manifolds according to some conformal invariants, which generalizes the conformally invariant sphere theorem of Chang-Gursky-Yang \cite{CGY}. Moreover, it provides a four-dimensional analogue of the well-known classification theorem of Schoen-Yau \cite{SY2} on 3-man…

We derive several mean value formulae on manifolds, generalizing the classical one for harmonic functions on Euclidean spaces as well as later results of Schoen-Yau, Michael-Simon, etc, on curved Riemannian manifolds. For the heat equation a mean value theorem with respect to `heat spheres' is proved for heat equation …

Doing surgery on the 5-torus, we construct a 5-dimensional closed spin-manifold M with $π_1(M) = Z^4times Z/3$, so that the index invariant in the KO-theory of the reduced $C^*$-algebra of $π_1(M)$ is zero. Then we use the theory of minimal surfaces of Schoen/Yau to show that this manifolds cannot carry a metric of pos…

A classical result of Sampson and Schoen-Yau in 1978 states that every diffeomorphism between compact hyperbolic Riemann surfaces is homotopic to an harmonic diffeomorphism. As conjectured by Schoen in 1993 and partially proved by Wan in 1992 and Tam-Wan in 1995, we prove in this article that this theorem generalizes t…

We prove the following comparison theorem for metrics with nonnegative scalar curvature, also known as the dihedral rigidity conjecture by Gromov: for $n\le 7$, if an $n$-dimensional prism has nonnegative scalar curvature and weakly mean convex faces, then its dihedral angle cannot be everywhere not larger than its Euc…

The study proves a rigidity theorem for compact manifolds with boundary.

We show that the periodic $η$-invariants introduced by Mrowka--Ruberman--Saveliev~\cite{MRS3} provide obstructions to the existence of cobordisms with positive scalar curvature metrics between manifolds of dimensions $4$ and $6$. The proof combines a relative version of the Schoen--Yau minimal surface technique with an…

We generalize classical theorems due to Lichnerowicz and Hitchin on the existence of Riemannian metrics of positive scalar curvature on spin manifolds to the case of foliated spin manifolds. As a consequence, we show that there is no foliation of positive leafwise scalar curvature on any torus, which generalizes the fa…

The study proves a new inequality and formula for manifolds with non-negative Ricci curvature.

Developing a singular dimension descent method for positive scalar curvature obstructions

Paper analyzes blowup of regularized Jang solutions and constant expansion surfaces.

In this paper, we show how to reduce the Penrose conjecture to the known Riemannian Penrose inequality case whenever certain geometrically motivated systems of equations can be solved. Whether or not these special systems of equations have general existence theories is therefore an important open problem. The key tool …

Free finite group actions on non-positively curved 3-manifolds

The paper proves conditions for the existence of small Urysohn width hypersurfaces in manifolds with positive scalar curvature.

Proves positive mass theorem for hyperbolic manifolds with ends.

In this paper, we focus on the geometry of compact conformally flat manifolds $(M^n,g)$ with positive scalar curvature. Schoen-Yau proved that its universal cover $(\widetilde{M^n},\tilde{g})$ is conformally embedded in $\mathbb{S}^n$ such that $M^n$ is a Kleinian manifold. Moreover, the limit set of the Kleinian group…

The study proves manifold properties related to positive scalar curvature.

Study of area minimizing surfaces in homotopy classes of maps.

Proves a higher-order positive energy theorem for stationary solutions in fourth-order gravity.

New non-perturbative counterexamples to Min-Oo's Conjecture are created.

We introduce a flow of maps from a compact surface of arbitrary genus to an arbitrary Riemannian manifold which has elements in common with both the harmonic map flow and the mean curvature flow, but is more effective at finding minimal surfaces. In the genus 0 case, our flow is just the harmonic map flow, and it tries…

We present a series of results concerning the interplay between the scalar curvature of a manifold and the mean curvature of its boundary. In particular, we give a complete topological characterization of those compact 3-manifolds that support Riemannian metrics of positive scalar curvature and mean-convex boundary and…

In this paper, we study the boundary behaviors of compact manifolds with nonnegative scalar curvature and with nonempty boundary. Using a general version of Positive Mass Theorem of Schoen-Yau and Witten, we prove the following theorem: For any compact manifold with boundary and nonnegative scalar curvature, if it is s…

Study disproves a generalized numerical criterion for certain pairs.

New polynomial criterion for periodic knots identified.

New algorithms optimize a soft-robust criterion in reinforcement learning, reducing conservatism.

The study explores positive scalar curvature metrics on non-orientable manifolds and their covers.

We introduce a new criterion to determine the order of an autoregressive model fitted to time series data. It has the benefits of the two well-known model selection techniques, the Akaike information criterion and the Bayesian information criterion. When the data is generated from a finite order autoregression, the Bay…

Paper introduces p-Laplace equations for curvature in conformal geometry.

Compact proof for Brown-York mass positivity and rigidity in flat and spherical spaces.

We develop a bubble tree construction and prove compactness results for $W^{2,2}$ branched conformal immersions of closed Riemann surfaces, with varying conformal structures whose limit may degenerate, in a compact Riemannian manifold with uniformly bounded areas and Willmore energies. The compactness property is appli…

Modified Bakry-Émery criterion inequality for Tsallis entropy monotonicity.

A widely applicable Bayesian information criterion (Watanabe, 2013) is applicable for both regular and singular models in the model selection problem. This criterion tends to overestimate the log marginal likelihood. We identify an overestimating term of a widely applicable Bayesian information criterion. Adjustment of…

Initial data with zero mass must be in pp-wave spacetimes.

New criterion improves predictive evaluation in weighted inference scenarios.