arXiv research

Proves Gerber statistic is always non-negative.

Proves positive mass theorem for 3-manifolds with a boundary.

New proof of Positive Mass Theorem using Green's function and monotonicity formula.

New proof removes decay assumptions for spacetime positive mass theorem.

New proofs given for space curves with totally positive torsion.

Simplified proof of cosmic singularity theorem using new mathematical techniques.

Let $(M,g)$ be a compact conformally flat manifold of dimension $n\geq4$ with positive scalar curvature. According to a positive mass theorem by Schoen and Yau, the constant term in the development of the Green function of the conformal Laplacian is positive if $(M,g)$ is not conformally equivalent to the sphere. On sp…

New proof shows spacetime energy is always positive in higher dimensions.

Let $(M,g)$ be a compact connected spin manifold of dimension $n\geq 3$ whose Yamabe invariant is positive. We assume that $(M,g)$ is locally conformally flat or that $n \in \{3,4,5\}$. According to a positive mass theorem of Witten, the constant term in the asymptotic development of the Green's function of the conform…

New proofs of unique photon surfaces in 4D spacetimes, extending previous work.

New proof shows equality in spacetime mass theorem.

In this paper, we develop a general study of contributions at infinity of Bochner-Weitzenböck-type formulas on asymptotically flat manifolds, inspired by Witten's proof of the positive mass theorem. As an application, we show that similar proofs can be obtained in a much more general setting as any choice of an irreduc…

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.

We derive a general obstruction to the existence of Riemannian metrics of positive scalar curvature on closed spin manifolds in terms of hypersurfaces of codimension two. The proof is based on coarse index theory for Dirac operators that are twisted with Hilbert C*-module bundles. Along the way we give a complete and s…

The paper provides a different proof of the result of Brendle-Schoen on the differential sphere theorem. It is shown directly that the invariant cone of curvature operators with positive (or non-negative) complex sectional curvature is preserved by the Ricci flow. This implies, by a result of Böhm-Wilking, that the nor…

This paper proves a conjecture about unique positive harmonic functions in a ball.

Proves Skoda's Division Theorem using degeneration and positivity of direct image bundles.

This paper is devoted, first of all, to give a complete unified proof of the Characterization Theorem for compact generalized $p-$Kähler manifolds (Theorem 3.2). The proof is based on the classical duality between "closed" positive forms and "exact" positive currents. In the last part of the paper we approach the gener…

The paper proves stability of the positive mass theorem using intrinsic flat convergence.

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

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

Alternative proof for non-existence of complete curves in differential strata.

Thurston's jiggling lemma simplifies triangulations.

New proof shows 4-manifolds can't support complex structures.

We extend the positive mass theorem proved previously by the author to the Lorentzian setting. This includes the original higher dimensional positive energy theorem whose spinor proof was given by Witten in dimension four and by Xiao Zhang in dimension five.

Proves positive mass theorem for specific manifold types.

New Sobolev inequalities on Kähler manifolds with positive Ricci curvature.

Quillen proved that repeated multiplication of the standard sesquilinear form to a positive Hermitian bihomogeneous polynomial eventually results in a sum of Hermitian squares, which was the first Hermitian analogue of Hilbert's seventeenth problem in the nondegenerate case. Later Catlin-D'Angelo generalized this posit…

New mass inequalities and proofs for causal variational principles.

In this article, the author provides full details of the proof of the concordance/isotopy problem. The first published proof, [5], accomplished this task only partially since there was an error, see the erratum [6], which damaged the main argument of [5, Theorem 2.9], and, consequently, the proof of [5, Theorem A].

New theorem for spacetime mass in noncompact regions.

The paper extends positivity results from vector bundles to Kobayashi positive ones.

Proves generic surjectivity of vector bundles via degeneration.

In this paper we study the Ricci flow on compact four-manifolds with positive isotropic curvature and with no essential incompressible space form. Our purpose is two-fold. One is to give a complete proof of Hamilton's classification theorem on four-manifolds with positive isotropic curvature and with no essential incom…

In this short note, using Siu-Yau's method [14], we give a new proof that any n-dimensional compact Kahler manifold with positive orthogonal bisectional curvature must be biholomorphic to $\mathbb{P}^n$.

We extend the Jang equation proof of the positive energy theorem due to R. Schoen and S.-T. Yau from dimension $n=3$ to dimensions $3 \leq n <8$. This requires us to address several technical difficulties that are not present when $n=3$. The regularity and decay assumptions for the initial data sets to which our argume…

We show that no exotic $\mathbb{R}^4$ admits a complete Riemannian metric with uniformly positive isotropic curvature and with bounded geometry. This is essentially a corollary of the main result in [Hu1], and was stated in [Hu2] without proof. In the process of the proof we also show that the diffeomorphism type of an…

Proves positive mass theorem for spin initial data sets with arbitrary ends and dominant energy shields.

We prove that under suitable assumptions, the constant term in the Green function of the Paneitz-Branson operator on a compact Riemannian manifold $(M,g)$ is positive unless $(M,g)$ is conformally diffeomophic to the standard sphere. The proof is inspired by the positive mass theorem on spin manifolds by Ammann-Humbert…

In this note we provide a proof of the following: Any compact KRS with positive bisectional curvature is biholomorphic to the complex projective space. As a corollary, we obtain an alternative proof of the Frankel conjecture by using the Kähler-Ricci flow.

New positive mass theorems for ALH manifolds with toroidal ends.

New proof of Haken's Lemma and Scharlemann's theorem using thin position.

Robot untangles knots by walking and switching crossings.

Generalising a proof by Bartnik in the asymptotically Euclidean case, we give an elementary proof of positivity of the hyperbolic mass near the hyperbolic space. It is a pleasure to dedicate this work to Robert Bartnik on the occasion of his 60th birthday.

Stability of positive mass theorem proven under Ricci curvature bounds.

In this short note, using an argument by Munteanu and Wang, we provide an alternative proof of the fact first obtained by Lei Ni that shrinking gradient Kähler-Ricci solitons with positive bisectional curvature must be compact.

On one hand, we study the class of graphs on surfaces, satisfying tessellation properties, with positive Forman curvature on each edge. Via medial graphs, we provide a new proof for the finiteness of the class, and give a complete classification. On the other hand, we classify the class of graphs on surfaces with posit…

Study determines scalar curvature invariants for 3-spheres embedded in 4-manifolds.