arXiv research

Proves inextendibility of weak null singularities from curvature blow-up.

Study weak super Ricci flow through neckpinch in metric measure spaces.

Paper builds singular metrics with constant Q-curvature.

Smooth flows with surgery approximate weak mean curvature flows with spherical and neck-pinch singularities.

The paper studies how singularities evolve in inverse mean curvature flow.

Estimates ends of Ricci shrinkers, focusing on smooth and singular cases.

Global weak solution found for harmonic map flow into Lorentzian manifolds.

Study on unique spacetime extensions in 1+1 dimensions with applications to weak null singularities.

We prove the existence and uniqueness of the weak Kahler-Ricci flow on projective varieties with log terminal singularities. It is also shown that the weak Kahler-Ricci flow can be uniquely continued through divisorial contractions and flips if they exist. We then propose an analytic version of the Minimal Model Progra…

In the category of metrics with conical singularities along a smooth divisor with angle in $(0, 2π)$, we show that locally defined weak solutions ($C^{1,1}-$solutions) to the Kähler-Einstein equations actually possess maximum regularity, which means the metrics are actually Hölder continuous in the singular polar coord…

In this paper, we prove that a sequence of weak almost Kähler-Ricci solitons under further suitable conditions converge to a Kähler-Ricci soliton with complex codimension of singularities at least 2 in the Gromov-Hausdorff topology. As a corollary, we show that on a Fano manifold with the modified K-energy bounded belo…

The paper proves a fibration theorem for collapsing sequences of Alexandrov spaces.

Consider the Kahler-Ricci flow with finite time singularities over any closed Kahler manifold. We prove the existence of the flow limit in the sense of current towards the time of singularity. This answers affirmatively a problem raised by Tian on the uniqueness of the weak limit from sequential convergence constructio…

New formulations for Ricci flows without smoothness.

In this paper we generalize the Local Removable Singularity Theorem in [16] for minimal laminations to the case of weak $H$-laminations (with $H\in \mathbb{R}$ constant) in a punctured ball of a Riemannian three-manifold. We also obtain a curvature estimate for any weak CMC foliation (with possibly varying constant mea…

Study restricts line arrangements with odd points using topological arguments.

New heat flow for harmonic maps avoids singularities but not bubbles.

Removes singularity order for Willmore immersions, reducing bubbling scenarios.

Using a method developped in [1] and [2], we prove the existence of weak non trivial solutions to fourth order elliptic equations with singularities and with critical Sobolev growth.

Symmetric graphs flow without singularities on their axis.

Weak Bianchi identity found for spacetimes with timelike singularities.

Paper reconciles different Ricci flow approaches and proves weak solutions.

Studying the behavior of the Kähler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Ampère equations. In this article, the third of a series on this subject, we study the long term behavior of the normalized Kähler-Ricci flow on mildly singula…

A new parametric method studies Willmore flows and energy quantization.

Proves weak cosmic censorship for a specific Einstein-scalar field system in 2+1 dimensions.

Studying the (long-term) behavior of the Kähler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Ampére equations. The purpose of this article, the second of a series on this subject, is to develop a viscosity theory for degenerate complex Mong…

Unified approach to stochastic Volterra systems' deviations.

We show that the Hawking--Penrose singularity theorem, and the generalisation of this theorem due to Galloway and Senovilla, continue to hold for Lorentzian metrics that are of $C^{1, 1}$-regularity. We formulate appropriate weak versions of the strong energy condition and genericity condition for $C^{1,1}$-metrics, an…

In this article, we consider the Angenent-Caputo-Knopf's Ricci Flow through neckpinch singularities. We will explain how one can see the A-C-K's Ricci flow through a neckpinch singularity as a flow of integral current spaces. We then prove the continuity of this weak flow with respect to the Sormani-Wenger Intrinsic Fl…

Constructs singular Yamabe solutions via equivariant reduction.

Minimax solutions are weak solutions to Cauchy problems involving Hamilton--Jacobi equations, constructed from generating families quadratic at infinity of their geometric solutions. We give a complete description of minimax solutions and we classify their generic singularities of codimension not greater than 2.

New findings show non-uniqueness in Ricci flow solutions for dimensions n≥5.

Study on naked singularities without symmetry, forming incomplete future null infinity and singular inner Cauchy horizon.

In the Compressed Sensing community, it is well known that given a matrix $X \in \mathbb R^{n\times p}$ with $\ell_2$ normalized columns, the Restricted Isometry Property (RIP) implies the Null Space Property (NSP). It is also well known that a small Coherence $μ$ implies a weak RIP, i.e. the singular values of $X_T$ l…

We prove the closure for the sequential weak $L^p$-topology of the class of vectorfields on $B^3$ having integer flux through almost every sphere. We show how this problem is connected to the study of the minimization problem for the Yang-Mills functional in dimension higher than critical, in the abelian case.

The paper studies weak singular Hermite-Einstein structures on homogeneous vector bundles.

Localizes smooth spaces to study their homotopy properties.

Smooth Yang-Mills fields proved in supercritical dimensions.

Weak base-point freeness leads to Kähler-Ricci flow diameter bounds.

Develops theory for Kähler-Ricci flow on singular varieties.

We study Jang's equation on a one-parameter family of asymptotically flat, spherically symmetric Cauchy hypersurfaces in the maximally extended Schwarzschild spacetime. The hypersurfaces contain apparent horizons and are parametrized by their proximity to the singularity at $r = 0$. We show that on those hypersurfaces …

Using the method of Nehari manifold, we prove the existence of at least two distinct weak solutions to elliptic equation of four order with singulatities and with critical Sobolev growth.

Study on Kähler-Einstein metrics on quasi-projective manifolds.

We define the Ricci curvature, as a measure, for certain singular torsion-free connections on the tangent bundle of a manifold. The definition uses an integral formula and vector-valued half-densities. We give relevant examples in which the Ricci measure can be computed. In the time dependent setting, we give a weak no…

Employing a notion of curvature for arbitrary closed sets we prove an ABP-type estimate for a class of singular submanifolds of arbitrary codimension and bounded mean curvature recently introduced by B. White. A weak-Harnack-type estimate is then derived using the ABP estimate. These results generalize analogous result…

A slice distance for the class of weak abelian Lp-bundles in 3 dimensions was introduced in a previous article in collaboration with Tristan Rivière, where it was used to prove the closure of such class of bundles for the weak Lp-convergence. We further investigate this distance here, and we prove more properties of it…

We classify the resolution graphs of weighted homogeneous surface singularities which admit rational homology disk smoothings. The nonexistence of rational homology disk smoothings is shown by symplectic geometric methods, while the existence is verified via smoothings of negative weights. In particular, it is shown th…

New framework for analyzing games with multi-dimensional singular controls and non-linear jumps.