arXiv research

Study real semi-stable degenerations and describe real loci via blow-ups.

Proves mass theorem for AF manifolds with conical singularities.

A steady state (or equilibrium point) of a dynamical system is hyperbolic if the Jacobian at the steady state has no eigenvalues with zero real parts. In this case, the linearized system does qualitatively capture the dynamics in a small neighborhood of the hyperbolic steady state. However, one is often forced to consi…

Study spinorial Yamabe problem on product manifolds, proving spike layer solutions.

Horizontal points of smooth submanifolds in stratified groups play the role of singular points with respect to the Carnot-Carathe'odory distance. When we consider hypersurfaces, they coincide with the well known characteristic points. In two step groups, we obtain pointwise estimates for the Riemannian surface measure …

We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half three sphere, by deforming conformally its standard metric. Using blow up analysis techniques and minimax arguments, we prove some existence and compactness results.

We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show …

Noise stabilizes solutions to transport equations, preventing blow-up.

Study on half spheres solves Nirenberg problem with complex blow-up analysis.

The paper defines and constructs almost complex blow-ups on 4D almost complex manifolds.

We present a simple uniqueness argument for a collection of McKean-Vlasov problems that have seen recent interest. Our first result shows that, in the weak feedback regime, there is global uniqueness for a very general class of random drivers. By weak feedback we mean the case where the contagion parameters are small e…

Formula derived for holomorphic Poisson blow-ups.

Study identifies numerical signs of blow-up in hydrodynamic equations.

We give several generalizations of the Kodaira vanishing and embedding theorems for Kähler manifolds to the case where the relevent line bundle has a small region of negative curvature. To prove the vanishing theorems we adapt techniques of Elworthy-Rosenberg for vanishing theorems in Riemannian geometry. For the embed…

Formula derived for Bott-Chern classes in complex blow-ups.

Explains blow-ups for Lie groupoids and algebroids, comparing different methods.

The paper derives a formula for Chow weights of toric blow-ups.

We find all intrinsic measures of $C^{1,1}$ smooth submanifolds in the Engel group, showing that they are equivalent to the corresponding $d$-dimensional spherical Hausdorff measure restricted to the submanifold. The integer $d$ is the degree of the submanifold. These results follow from a different approach to negligi…

Study on curvature blow-up rates in black hole interiors from gravitational collapse.

The paper examines the blow-up of Ricci curvatures in conformal metrics.

We show that the blow-up of a generalized Kahler 4-manifold in a nondegenerate complex point admits a generalized Kahler metric. As with the blow-up of complex surfaces, this metric may be chosen to coincide with the original outside a tubular neighbourhood of the exceptional divisor. To accomplish this, we develop a b…

This paper studies a specific blow-up algorithm for sop polynomials and their RLCT.

We study blow-ups in generalized complex geometry. To that end we introduce the concept of holomorphic ideal, which allows one to define a blow-up in the category of smooth manifolds. We then investigate which generalized complex submanifolds are suitable for blowing up. Two classes naturally appear; generalized Poisso…

Proves mass theorem up to dimension 19 using symmetrization and singularity techniques.

The paper analyzes high-dimensional sphere solutions to the Nirenberg problem with residual mass.

The generalized Jang equation was introduced in an attempt to prove the Penrose inequality in the setting of general initial data for the Einstein equations. In this paper we give an extensive study of this equation, proving existence, regularity, and blow-up results. In particular, precise asymptotics for the blow-up …

The Yamabe flow can blow up in infinite time with small perturbations.

In this paper, we study the blow-up of a locally conformal symplectic manifold.We show that there exists a locally conformal symplectic structure on the blow-up of a locally conformal symplectic manifold along a compact induced symplectic submanifold.

Study on spinor field equation on spheres, focusing on blow-up analysis.

Two approaches study the homotopy of blow ups in algebraic and symplectic geometry.

Study on blow-up behavior of sign-changing solutions for Yamabe equation.

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

Desingularizes singular foliations with a locally compact groupoid.

Corrected Monti's blow-up analysis for H-minimizing sets in Heisenberg group.

We prove that tangent cones to 2-dimensional calibrated cycles are unique. Using this result we prove a rate of convergence for the mass of the blow-up of a calibrated integral 2-cycle towards the limiting density. With the same techniques, we can also prove such a rate for J-holomorphic maps between almost complex man…

The article proves conditions for blow-ups of lcK spaces to remain lcK.

Finite-time blow-up in Yang-Mills flow for small energy initial connections.

The aim of this article is to study expansions of solutions to an extremal metric type equation on the blow-up of constant scalar curvature Kähler surfaces. This is related to finding constant scalar curvature Kähler (cscK) metrics on K-stable blow-ups of extremal Kähler surfaces

Study shows continuity and geometric regularity of Kähler-Ricci flow blow-up limits.

We derive a blow-up formula for the de Rham cohomology of a local system of complex vector spaces on a compact complex manifold. As an application, we obtain the blow-up invariance of $E_{1}$-degeneracy of the Hodge-de Rham spectral sequence associated to a local system of complex vector spaces.

We study the Yang-Mills flow on a holomorphic vector bundle E over a compact Kahler manifold X. We construct a natural barrier function along the flow, and introduce some techniques to study the blow-up of the curvature along the flow. Making some technical assumptions, we show how our techniques can be used to prove t…

We show that at generic points blow-ups/tangents of differentiability spaces are still differentiability spaces; this implies that an analytic condition introduced by Keith as an inequality (and later proved to actually be an equality) passes to tangents. As an application, we characterize the $p$-weak gradient on iter…

We continue the study of blow-ups in generalized complex geometry with the blow-up theory for generalized Kähler manifolds. The natural candidates for submanifolds to be blown-up are those which are generalized Poisson for one of the two generalized complex structures and can be blown up in a generalized complex manner…

We establish blow-up profiles for any blowing-up sequence of solutions of general conformally invariant fully nonlinear elliptic equations on Euclidean domains. We prove that (i) the distance between blow-up points is bounded from below by a universal positive number, (ii) the solutions are very close to a single stand…

New boundary condition for weak inverse mean curvature flow in bounded domains.

CR structure on S³ with non-compact solutions to CR Yamabe problem.

Study on curvature blow-up and convergence of continuity method on Hirzebruch surface.

We study blow-ups around fixed points at Type I singularities of the Ricci flow on closed manifolds using Perelman's W-functional. First, we give an alternative proof of the result obtained by Naber and Enders-Müller-Topping that blow-up limits are non-flat gradient shrinking Ricci solitons. Our second and main result …