arXiv research

Paper solves degenerated circle pattern metric problem in spherical geometry.

The paper proves rigidity for warped product spaces with degenerate ends.

Study calculates curvatures in holomorphic fibrations using degenerate Hermitian forms.

A diffeomorphism of pseudo-Riemannian manifolds is called sectional curvature preserving if it preserves the sectional curvature of all the nondegenerate 2-planes. We consider a similar condition for degenerate 2-planes and we prove that the diffeomorphism is conformal (when the condition is fulfilled for weakly degene…

New proof shows certain 4D metrics are non-degenerate if curvature is negative definite.

In infinite dimensional Heisenberg group, degenerate distances linked to unbounded curvature.

Smooth solutions up to evolving free boundaries for degenerate equations.

Paper studies degenerated circle packings in hyperbolic geometry and finds conditions for their existence.

Integrability of mean curvature near degenerate points in Heisenberg group.

The study shows that nonpositively curved 4-manifolds with zero Euler characteristic have degenerating Ricci curvature.

Study non-degenerate anisocurved surfaces in homogeneous 3-manifolds.

A mixed type surface is a connected regular surface in a Lorentzian 3-manifold with non-empty spacelike and timelike point sets. The induced metric of a mixed type surface is a signature-changing metric, and their lightlike points may be regarded as singular points of such metrics. In this paper, we investigate the beh…

In our earlier articles we studied tube hypersurfaces in ${\mathbb C}^3$ that are 2-nondegenerate and uniformly Levi degenerate of rank 1. In particular, we showed that the vanishing of the CR-curvature of such a hypersurface is equivalent to the Monge equation with respect to one of the variables. In the present paper…

This is a continuation of a previous paper of same title. The degeneration, i.e. curvature blow-up, of sequences of metrics appoaching the Sigma constant, assumed non-positive, is analysed. The degeneration is related to the sphere decomposition of the 3-manifold M, in case M is sigma-tame.

Study bifurcations of curves on surfaces in Minkowski 3-space.

This paper analyses the convergence and degeneration of sequences of metrics on a 3-manifold, and relations of such with Thurston's geometrization conjecture. The sequences are minimizing sequences for a certain (optimal) scalar-curvature type functional and their degeneration is related to the sphere and torus decompo…

K-polystability of a polarised variety is an algebro-geometric notion conjecturally equivalent to the existence of a constant scalar curvature Kähler metric. When a variety is K-unstable, it is expected to admit a "most destabilising" degeneration. In this note we show that if such a degeneration exists, then the limit…

Geodesics in Kähler metrics connect metrics with constant scalar curvature.

The paper studies connections in superintegrable systems, revealing geometric insights.

We continue our study, initiated in an earlier article, of a class of rigid hypersurfaces in ${\mathbb C}^3$ that are 2-nondegenerate and uniformly Levi degenerate of rank 1, having zero CR-curvature. We drop the restrictive assumptions of the earlier paper and give a complete description of the class. Surprisingly, th…

A new spinorial heat flow framework studies geometric degeneration on 3-manifolds.

We study a generalization of constant Gauss curvature -1 surfaces in Euclidean 3-space, based on Lorentzian harmonic maps, that we call pseudospherical frontals. We analyze the singularities of these surfaces, dividing them into those of characteristic and non-characteristic type. We give methods for constructing all n…

A spacelike surface in four-dimensional Lorentz-Minkowski spacetime through the lightcone has a meaningful lightlike normal vector field $η$. Several sufficient assumptions on such a surface with non-degenerate $η$-second fundamental form are established to prove that it must be a totally umbilical round sphere. With t…

We study singularities of Gauss maps of fronts and give characterizations of types of singularities of Gauss maps by geometric properties of fronts which are related to behavior of bounded principal curvatures. Moreover, we investigate relation between a kind of boundedness of Gaussian curvatures near cuspidal edges an…

We consider the class of locally boost isotropic spacetimes in arbitrary dimension. For any spacetime with boost isotropy, the corresponding curvature tensor and all of its covariant derivatives must be simultaneously of alignment type ${\bf D}$ relative to some common null frame. Such spacetimes are known as type ${\b…

We construct almost complex algebraic curvature tensors for pseudo Hermitian inner products whose skew-symmetric curvature operator has constant Jordan normal form on the set of non-degenerate complex lines.

The paper discusses polynomial convergence to conical Kähler-Einstein metrics.

This paper mainly aims to establish the well-posedness on time interval $[0,\varepsilon^{-\frac{1}{2}}T]$ of the classical initial problem for the bosonic membrane in the light cone gauge. Here $\varepsilon$ is the small parameter measures the nonlinear effects. In geometric, the bosonic membrane are timelike submanifo…

This paper is a continuation of I, (same title), and is concerned with the existence, regularity and degeneration of metrics minimizing natural curvature functionals on the space of metrics on 3-manifolds. The functionals chosen are designed to be optimal w.r.t. the issue of geometrization of the underlying 3-manifold,…

In this paper we prove non-existence and classification results for elliptic fully nonlinear elliptic degenerate conformal equations on certain subdomains of the sphere with prescribed constant mean curvature along its boundary. We also consider non-degenerate equations. Such subdomains are the hemisphere (or a geodesi…

Study existence of harmonic and Dirac-harmonic maps from degenerating surfaces.

The classical Cartan's structural equations show in a compact way the relation between a connection and its curvature, and reveals their geometric interpretation in terms of moving frames. In order to study the mathematical properties of singularities, we need to study the geometry of manifolds endowed on the tangent b…

The paper studies Möbius inversion on surfaces in Minkowski 3-space.

Classifies surfaces translating under specific curvature flows.

In this paper, we study the existence of various harmonic maps from Hermitian manifolds to Kaehler, Hermitian and Riemannian manifolds respectively. By using refined Bochner formulas on Hermitian (possibly non-Kaehler) manifolds, we derive new rigidity results on Hermitian harmonic maps from compact Hermitian manifolds…

We study Yamabe metrics, and the moduli space of Yamabe metrics, on an arbitrary closed 3-manifold M. The main focus is on the boundary behavior of the moduli space, i.e. the behavior of degenerating sequences of unit volume Yamabe metrics on M. It is proved that such degenerations, when non-trivial in a certain sense,…

Compactness theory for biharmonic maps on degenerating Einstein manifolds.

We note that the Bogomolny equation for abelian vortices is precisely the condition for invariance of the Hermitian-Einstein equation under a degenerate conformal transformation. This leads to a natural interpretation of vortices as degenerate hermitian metrics that satisfy a certain curvature equation. Using this view…

Study on positive solutions of Yamabe-type equation on spheres.

Paper develops estimates for Lagrangian phase changes in 2D.

Study behavior of curvatures near singular points of frontals.

The study finds multiple maxima for eigenfunctions on positively curved spheres.

We prove conjectures of Rene Thom and Vladimir Arnold for C^2 solutions to the degenerate elliptic equation that is the level set equation for motion by mean curvature. We believe these results are the first instances of a general principle: Solutions of many degenerate equations behave as if they are analytic, even wh…

Proves existence of Yang-Mills fields for specific curvature conditions.

Study focal surfaces of wave fronts with unbounded curvatures.

We study critical Riemannian 4-manifolds with a lower bound on Ricci curvature, but no a priori analytic constraints such as on Sobolev constants. We derive elliptic-type estimates for the local curvature radius, which itself controls sectional curvature. The primary method is construction of blow-ups of degenerating m…

Flow on weighted graphs sharpens Bakry-Émery curvature.

In this note, we study Q-curvature flow on $S^4$ with indefinite nonlinearity. Our result is that the prescribed Q-curvature problem on $S^4$ has a solution provided the prescribed Q-curvature $f$ has its positive part, which possesses non-degenerate critical points such that $Δ_{S^4} f\not=0$ at the saddle points and …