arXiv research

Estimates Bartnik mass for metrics with nonnegative Gauss curvature.

We study different notions of Riemannian curvatures: The $p$-curvatures which interpolate between the scalar curvature and the sectional curvature, the Gauss-Bonnet-Weyl curvatures form another interpolation from the scalar curvature to the Gauss-Bonnet integrand. We bring out the $(p,q)$-curvatures, which incorporate …

We study the evolution of convex complete non-compact graphs by positive powers of Gauss curvature. We show that if the initial complete graph has a local uniform convexity, then the graph evolves by any positive power of Gauss curvature for all time. In particular, the initial graph is not necessarily differentiable.

Authors prove a formula relating the Gaussian curvature of polyhedral vertex stars to their Gauss images.

Study examines noncompact cases of Gauss Curvature Flow on revolution surfaces.

As an interesting application of the Einstein-Gauss-Bonnet theory and our work on the Gauss-Bonnet-Chern mass (Ge, Wang, Wu), we obtain a positive mass theorem for asymptotically flat graphs in $\R^{n+1}$ under a condition that $R+αL_2$ is non-negative, where $R$ is the scalar curvature, $α\in\R$ a constant and $L_2$ t…

Study relates Gaussian curvature signs to cuspidal edge types and geometric invariants.

We extend our previous definition of quasi-local mass to 2-spheres whose Gauss curvature is negative and prove its positivity.

Classification of constant curvature surfaces in Berger spheres.

New geometric inequality for mass from immersed submanifolds.

Flow adjusts curvature to avoid a fixed region, proving bounds and regularity.

In this paper we extend Efimov's Theorem by proving that any complete surface in $\mathbb{R}^3$ with Gauss curvature bounded above by a negative constant outside a compact set has finite total curvature, finite area and is properly immersed. Moreover, its ends must be asymptotic to half-lines. We also give a partial so…

Study on anisotropic curvature flow for noncompact convex hypersurfaces.

We are concerned with spacelike convex hypersurfaces of positive constant (K-hypersurfaces) or prescribed Gauss curvature in Minkowski space. Our main purpose is to study entire solutions as well as the Dirichlet problem in bounded domains of the related Monge-Ampere equation.

Paper bounds total geodesic curvature using boundary data in hyperbolic gravity.

In this article, we give the integrability conditions for the existence of an isometric immersion from an orientable simply connected surface having prescribed Gauss map and positive extrinsic curvature into some unimodular Lie groups. In particular, we discuss the case when the Lie group is the euclidean unit sphere $…

We establish a nice orthonormal frame field on a closed surface minimally immersed in a unit sphere $S^{n}$, under which the shape operators take very simple forms. Using this frame field, we obtain an interesting property $K+K^{N}=1$ for the Gauss curvature $K$ and the normal curvature $K^{N}$ if the Gauss curvature i…

In this paper, we study the geometry of the manifolds of geodesics of a Zoll surface of positive Gauss curvature, show how these metrics induce Finsler metrics of constant flag curvature and give some explicit constructions.

Paper proves biharmonic hypersurfaces in nonzero space form have constant mean curvature.

The paper studies how convex hypersurfaces in hyperbolic space evolve under a specific curvature flow.

Convex hypersurfaces in hyperbolic space evolve to geodesic spheres.

Study surfaces with nonvanishing Gauss curvature in Lorentz-Minkowski space.

Canonical principal parameters are introduced for surfaces in $\mathbb R^3$ without umbilical points. It is proved that in these parameters the surface is determined (up to position in space) by a pair of invariants satisfying a partial differential equation equivalent to the Gauss equation. As such a pair of invariant…

The $c$-curvature of a complete surface with Gauss curvature close to 1 in $C^2$ norm is almost-positive (in the sense of Kim--McCann). Our proof goes by a careful case by case analysis combined with perturbation arguments from the constant curvature case, keeping track of an estimate on the closeness curvature conditi…

Flow of convex hypersurfaces in hyperbolic space converges to geodesic spheres.

The paper improves the Gauss curvature estimation for harmonic surfaces and verifies a modified defect relation.

Sharp lower bound for curvature in Kähler manifolds.

We prove a general contractibility criterion for Riemannian metrics on a disc.

In this paper we study constant positive Gauss curvature $K$ surfaces in the 3-sphere $S^3$ with $0<K<1$ as well as constant negative curvature surfaces. We show that the so-called normal Gauss map for a surface in $S^3$ with Gauss curvature $K<1$ is Lorentz harmonic with respect to the metric induced by the second fun…

This paper studies non-compact Ricci surfaces with catenoidal ends.

Constructs extensions for Bartnik data to approach mass limits.

The paper studies surface contraction in hyperbolic and spherical spaces.

Classifies surfaces translating under specific curvature flows.

Applying a local Gauss-Bonnet formula for closed subanalytic sets to the complex analytic case, we obtain characterizations of the Euler obstruction of a complex analytic germ in terms of the Lipschitz-Killing curvatures and the Chern forms of its regular part. We also prove analogous results for the global Euler obstr…

In this paper we introduce a mass for asymptotically flat manifolds by using the Gauss-Bonnet curvature. We first prove that the mass is well-defined and is a geometric invariant, if the Gauss-Bonnet curvature is integrable and the decay order $τ$ satisfies $τ> \frac {n-4}{3}.$ Then we show a positive mass theorem for …

We generalise the classical Chern-Gauss-Bonnet formula to a class of 4-dimensional manifolds with finitely many conformally flat ends and singular points. This extends results of Chang-Qing-Yang in the smooth case. Under the assumptions of finite total Q curvature and positive scalar curvature at the ends and at the si…

We characterize Bianchi-Bäcklund transformations of surfaces of positive constant Gauss curvature in terms of dressing actions of the simplest type on the space of harmonic maps.

Unified interpretation of sub-Riemannian Gauss-Bonnet theorem for surfaces in 3D contact manifolds.

The paper is centered around a new proof of the infinitesimal rigidity of smooth closed surfaces with everywhere positive Gauss curvature. We use a reformulation that replaces deformation of an embedding by deformation of the metric inside the body bounded by the surface. The proof is obtained by studying derivatives o…

This is a paper based on a talk given at the conference on Conformal Geometry which held at Roscoff in France in the 2008 summer. We study some aspects of the equation arising from the problem of the existence on a given closed Riemannian manifold of dimension at leat 4, of a conformal metric with constant $h_4$ curvat…

The paper studies how spacelike surfaces evolve in Lorentz-Minkowski space over time.

The second H. Weyl curvature invariant of a Riemannian manifold, denoted $h_4$, is the second curvature invariant which appears in the well known tube formula of H. Weyl. It coincides with the Gauss-Bonnet integrand in dimension 4. A crucial property of $h_4$ is that it is nonnegative for Einstein manifolds, hence it p…

The paper proves simplicial volume positivity for certain nonpositively curved 4-manifolds with nonzero Euler characteristic.

Study on rotational hypersurfaces with constant Gauss-Kronecker curvature.

Study rotational surfaces with prescribed Gauss curvature in 3D space.

Paper solves Orlicz-Aleksandrov problem using Gauss curvature flow.

Integral geometry explores curvature conjectures on manifolds.

Study surfaces with nonnegative curvature in spectral sense, proving inequalities and bounds.