arXiv research

This paper classifies Legendre singularities of sub-Riemannian geodesics on surfaces.

Classifies 4D toric Hermitian ALF metrics with conical singularities.

The study examines conditions for scalar curvature and Dirac operators on singular spaces.

Smooths metrics with nonnegative scalar curvature near singular sets.

Investigates singular Finsler foliations on $(α,β)$-spaces and their relation to Riemannian foliations.

Proves conditions for positive scalar curvature on certain manifolds with conical singularities.

The study examines Riemannian surfaces with simple singularities.

Locally, isoperimetric problems on Riemannian surfaces are sub-Riemannian problems in dimension 3. The particular case of Dido problems corresponds to a class of singular contact sub-Riemannian metrics : metrics which have the charateristic vertor field as symmetry. We give a classification of the generic conjugate loc…

On a Riemannian or a semi-Riemannian manifold, the metric determines invariants like the Levi-Civita connection and the Riemann curvature. If the metric becomes degenerate (as in singular semi-Riemannian geometry), these constructions no longer work, because they are based on the inverse of the metric, and on related o…

Constructs ancient solutions to mean curvature flow with prescribed singular sets.

Prove Riemannian positive mass theorem for asymptotically flat metrics with low-codimension singularities.

Smooth metrics satisfying Penrose inequality are necessarily smooth.

The paper proves local isometric embeddings for singular metrics near a point.

The paper studies affine connections on singular warped products and their curvature.

Abstract: Proves no non-trivial normal orbits for specific Hamiltonians.

Study finds conditions for existence of specific pseudo-Riemannian cobordisms.

Proves finite step termination of Kähler-Einstein metric singularity formation.

The Schwarzschild spacetime metric of negative mass is well-known to contain a naked singularity. In a spacelike slice, this singularity of the metric is characterized by the property that nearby surfaces have arbitrarily small area. We develop a theory of such "zero area singularities" in Riemannian manifolds, general…

It is proved by Kawamata that the canonical bundle of a projective manifold is semi-ample if it is big and nef. We give an analytic proof using the Ricci flow, degeneration of Riemannian manifolds and $L^2$-theory. Combined with our earlier results, we construct unique singular Kahler-Einstein metrics with a global Rie…

Given $(M,g_0)$ a closed Riemannian manifold and a nonempty closed subset $X$ in $M$, the singular $σ_k-$Yamabe problem asks for a complete metric $g$ on $M\backslash X$ conformal to $g_0$ with constant $σ_k-$curvature. The $σ_k-$curvature is defined as the $k-$th elementary symmetric function of the eigenvalues of the…

Study on metrics with positive scalar curvature on manifolds with singularities.

The connected components of the zero set of any conformal vector field, in a pseudo-Riemannian manifold of arbitrary signature, are shown to be totally umbilical conifold varieties, that is, smooth submanifolds except possibly for some quadric singularities. The singularities occur only when the metric is indefinite, i…

Smooth bundles with rough data maintain Hodge kernel isomorphism.

A compact manifold is called Bieberbach if it carries a flat Riemannian metric. Bieberbach manifolds satisfy an isosystolic inequality by a general and fundamental result of M. Gromov. In dimension 3, there exist four classes of non-orientable Bieberbach manifolds up to an affine diffeomorphism. In this paper, We prove…

Global singularities propagate in magnetic mechanical systems on Riemannian manifolds.

We prove a compactness theorem for metrics with Bounded Integral Curvature on a fixed closed surface $Σ$. As a corollary, we obtain a compactification of the space of Riemannian metrics with conical singularities, where an accumulation of singularities is allowed.

Positive curvature forces foliation leaf spaces to have boundaries.

In this paper we prove local existence of a Ricci de Turck flow starting at a space with incomplete edge singularities and flowing for a short time within a class of incomplete edge manifolds. We derive regularity properties for the corresponding family of Riemannian metrics and discuss boundedness of the Ricci curvatu…

We consider the quantum completeness problem, i.e. the problem of confining quantum particles, on a non-complete Riemannian manifold $M$ equipped with a smooth measure $ω$, possibly degenerate or singular near the metric boundary of $M$, and in presence of a real-valued potential $V\in L^2_{\mathrm{loc}}(M)$. The main …

For a Riemannian foliation F on a compact manifold M , J. A. Álvarez López proved that the geometrical tautness of F , that is, the existence of a Riemannian metric making all the leaves minimal submanifolds of M, can be characterized by the vanishing of a basic cohomology class (the Álvarez class). In this work we gen…

We analyze sub-Riemannian and lightlike metrics from the point of view of their rigidity as geometric structures. Following Cartan's and Gromov's formal definitions, they are never rigid, yet, in generic cases, they naturally give rise to rigid geometric structures!?

We present a survey on generic singularities of geodesic flows in smooth signature changing metrics (often called pseudo-Riemannian) in dimension 2. Generically, a pseudo-Riemannian metric on a 2-manifold $S$ changes its signature (degenerates) along a curve $S_0$, which locally separates $S$ into a Riemannian ($R$) an…

Combining several previously known arguments, we prove marked length spectrum rigidity for surfaces with nonpositively curved Riemannian metrics away from a finite set of cone-type singularities with cone angles $>2π$. With an additional condition, we can weaken the requirement on one metric to `no conjugate points.'

A 3D almost-Riemannian manifold is a generalized Riemannian manifold defined locally by 3 vector fields that play the role of an orthonormal frame, but could become collinear on some set $\Zz$ called the singular set. Under the Hormander condition, a 3D almost-Riemannian structure still has a metric space structure, wh…

We consider an inverse problem associated with some 2-dimensional non-compact surfaces with conical singularities, cusps and regular ends. Our motivating example is a Riemann surface $\mathcal M = Γ\backslash{\bf H}^2$ associated with a Fuchsian group of the 1st kind $Γ$ containing parabolic elements. $\mathcal M$ is t…

The paper studies metrics with constant scalar curvature on foliated manifolds.

We study the question of existence of a Riemannian metric of positive scalar curvature metric on manifolds with the Sullivan-Baas singularities. The manifolds we consider are Spin and simply connected. We prove an analogue of the Gromov-Lawson Conjecture for such manifolds in the case of particular type of singularitie…

Study on Sol's metric spheres and their properties.

Existence of non-trivial monopoles on 3-spheres proven.

We consider a pseudo-Riemannian metric that changes signature along a smooth curve on a surface, called the discriminant curve. The discriminant curve separates the surface locally into a Riemannian and a Lorentzian domain. We study the local behaviour and properties of geodesics at a point on the discriminant where th…

We obtain a compactness result for various classes of Riemannian metrics in dimension four; in particular our method applies to anti-self-dual metrics, Kahler metrics with constant scalar curvature, and metrics with harmonic curvature. With certain geometric assumptions, the moduli space can be compactified by adding m…

We study positive scalar curvature on the regular part of Riemannian manifolds with singular, uniformly Euclidean ($L^\infty$) metrics that consolidate Gromov's scalar curvature polyhedral comparison theory and edge metrics that appear in the study of Einstein manifolds. We show that, in all dimensions, edge singularit…

Building on author's previous results in singular semi-Riemannian geometry and singular general relativity, the behavior of gauge theory at singularities is analyzed. The usual formulations of the field equations at singularities are accompanied by infinities which block the evolution equations, mainly because the metr…

Geodesics near singularities either hit or wind around, with winding number dependent on singularity type.

Formula for heat coefficients of curved conic singularities derived from Riemannian metrics.

The study shows that certain metrics on spheres prevent stable tangent cones for area-minimizing boundaries.

Conic singular sub-manifolds are Lipschitz Normally Embedded in compact non-Euclidean manifolds.

Motivated by the use of degenerate Jacobi metrics for the study of brake orbits and homoclinics, we develop a Morse theory for geodesics in conformal metrics having conformal factors vanishing on a regular hypersurface of a Riemannian manifold.