arXiv research

New extremal Poincaré type metrics found via blow-up theorem.

Proves a generalized vanishing theorem for quasi-smooth stacks, with applications in K-theory and birational geometry.

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…

We prove a theorem of Leray-Hirsch type and give an explicit blow-up formula for Dolbeault cohomology on (\emph{not necessarily compact}) complex manifolds. We give applications to strongly $q$-complete manifolds and the $\partial\bar{\partial}$-lemma.

Proves mass theorem for AF manifolds with conical singularities.

Ancient solutions arise in the study of parabolic blow-ups. If we can categorize ancient solutions, we can better understand blow-up limits. Based on an argument of Giga and Kohn, we give a Liouville-type theorem restricting ancient, type-I, non-collapsing two- dimensional mean curvature flows to either spheres or cyli…

The paper proves rigidity theorems for Type II singularities in Lagrangian flows.

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

We use a sheaf-theoretic approach to obtain a blow-up formula for Dolbeault cohomology groups with values in the holomorphic vector bundle over a compact complex manifold. As applications, we present several positive (or negative) examples associated to the vanishing theorems of Girbau, Kawamata-Viehweg and Green-Lazar…

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

For a sequence of blow up solutions of the Yamabe equation on non-locally confonformally flat compact Riemannian manifolds of dimension 10 or 11, we establish sharp estimates on its asymptotic profile near blow up points as well as sharp decay estimates of the Weyl tensor and its covariant derivatives at blow up points…

Using results of Gathmann, we prove the following theorem: If a smooth projective variety X has generically semisimple (p,p)-quantum cohomology, then the same is true for the blow-up of X at any number of points. This a successful test for a modified version of Dubrovin's conjecture from the ICM 1998.

The blow-up rates of derivatives of the curvature function will be presented when the closed curves contract to a point in finite time under the general curve shortening flow. In particular, this generalizes a theorem of M.E. Gage and R.S. Hamilton about mean curvature flow in $\mathbb{R}^{2}$.

Defines a new calculus for cusp pseudodifferential operators and proves index theorems.

Blowing up flat metrics yields balanced ones with constant curvature.

Formula derived for blow-up of quaternionic maps on Hyperkähler manifolds.

The paper defines a critical value for a magnetic system and extends solutions beyond blow-up.

Utilizing a splitting of geometric flows on surfaces introduced by Buzano and Rupflin, we present a general scheme to prove blow up criteria for such geometric flows. A vital ingredient is a new compactness theorem for families of metrics on surfaces with a uniform bound on their volumes, square integrals of their curv…

Proves existence of Yamabe metrics on conical 4-manifolds using min-max method.

In this note we establish several versions of a compactness theorem for submanifolds. In particular we require only bounds on the second fundamental form and do not assume volume or diameter bounds. As an application we prove a compactness theorem for mean curvature flows and use it to construct smooth blow-up limits a…

The paper studies how submanifolds in Gaussian space behave under mean curvature flow, showing they typically blow up.

Proves existence and compactness of solutions to $σ_2$-Nirenberg problem on sphere.

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…

The weight $θ$-sheaf $\underline{\mathbb{R}}_{X,θ}$ helps us to reinterpret Morse-Novikov cohomologies via sheaf theory. We give several theorems of Künneth and Leray-Hirsch types. As applications, we prove that the $θ$-Lefschetz number is independent of $θ$ and calculate the Morse-Novikov cohomologies of projective bu…

New estimates quantify blow-up rates of spacelike singularities in gravitational collapse.

We study compactness for nonnegative solutions of the fourth order constant $Q$-curvature equations on smooth compact Riemannian manifolds of dimension $\ge 5$. If the $Q$-curvature equals $-1$, we prove that all solutions are universally bounded. If the $Q$-curvature is $1$, assuming that Paneitz operator's kernel is …

In this paper, we prove that the mean curvature blows up at the same rate as the second fundamental form at the first singular time $T$ of any compact, Type I mean curvature flow. For the mean curvature flow of surfaces, we obtain similar result provided that the Gaussian density is less than two. Our proofs are based …

Paper uses 3-circle theorem to study Willmore surfaces and prove decay estimates.

Compactness proven for specific dimensions of manifolds with boundary conditions.

On a compact complex manifold we study the behaviour of strong Kähler with torsion (strong KT) structures under small deformations of the complex structure and the problem of extension of a strong KT metric. In this context we obtain the analogous result of Miyaoka extension theorem. Studying the blow-up of a strong KT…

Sharp rates and symmetry for higher order conformally invariant equations near singularities.

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

Proves regularity for stable varifolds near specific cones.

We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measu…

New, shorter proofs for varifolds and flows with improved decay of flatness.

Study of separatrix configurations in holomorphic flows with real time.

Study on hyperbolic elastic flow, proving convergence and quantifying singularities.

Extends Campanato theory to multi-valued functions for geometric variational problems.

Kontsevich's classes distinguish smooth structures on fiber bundles.

In this paper, we introduce and study the conformal mean curvature flow of submanifolds of higher codimension in the Euclidean space $\bbr^n$. This kind of flow is a special case of a general modified mean curvature flow which is of various origination. As the main result, we prove a blow-up theorem concluding that, un…

Curves become nearly circular over time without initial assumptions.

Holomorphic Koszul-Brylinski homology studied via Dolbeault cohomology.

We give a new proof of Brakke's partial regularity theorem up to C^{1,ς} for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is the sum of the mean curvature and an…

The paper analyzes finite-time singularities in Spin(7)-structure flows using Shi-type estimates.

We prove an existence theorem for Asymptotically Conical Ricci Flat Kahler metrics in $\mathbb{C}^2$ with cone singularities along a smooth complex curve. These metrics are expected to arise as blow up limits of non collapsed sequences of Kahler Einstein metrics with cone singularities.

In this article, I prove the following statement: Every compact complex surface with even first Betti number is deformation equivalent to one which admits an extremal Kähler metric. In fact, this extremal Kähler metric can even be taken to have constant scalar curvature in all but two cases: the deformation equivalence…

Study spherical metrics on flat torus with cone singularities.

Extends positive mass theorem to arbitrary dimensions using a new inductive scheme.