arXiv research

Two-dimensional collapsed spaces with lower Ricci bounds are topological surfaces.

Special Lagrangian submanifolds emerge from K3 surface collapse.

Study on K3 surfaces' collapsing and special Kähler structures.

Study on energy of maps from K3 surface to flat orbifold.

Study describes limits of non-collapsing K3 surfaces using algebraic data.

We provide a quantitative obstruction to collapsing surfaces of genus at least 2 under a lower curvature bound and an upper diameter bound. Keywords: curvature; diameter; volume; filling radius; systole; Gromov-Hausdorff distance

Smoothness of collapsed regions in soap films is proven, indicating wetted singularities.

Soap films collapse only if their bulk has negative pressure, forming convex shapes.

We exhibit families of Ricci-flat Kahler metrics on K3 surfaces which collapse to an interval, with Tian-Yau and Taub-NUT metrics occurring as bubbles. There is a corresponding continuous surjective map from the K3 surface to the interval, with regular fibers diffeomorphic to either 3-tori or Heisenberg nilmanifolds.

We study the mean curvature flow of hypersurfaces in $\R^{n+1}$, with initial surfaces sufficiently close to the standard $n$-dimensional sphere. The closeness is in the Sobolev norm with the index greater than $\frac{n}{2}+1$ and therefore it does not impose restrictions of the mean curvature of the initial surface. W…

Researchers create initial data for multiple collapsing boson stars.

The Chern-Ricci flow is an evolution equation of Hermitian metrics by their Chern-Ricci form, first introduced by Gill. Building on our previous work, we investigate this flow on complex surfaces. We establish new estimates in the case of finite time non-collapsing, anologous to some known results for the Kahler-Ricci …

The torus cannot collapse to a segment under certain curvature conditions.

We provide a moduli-theoretic framework for the collapsing of Ricci-flat Kahler metrics via compactification of moduli varieties of Morgan-Shalen and Satake type. In patricular, we use it to study the Gromov-Hausdorff limits of hyperKahler metrics with fixed diameters, especially for K3 surfaces.

We consider the mean curvature evolution of rotationally symmetric surfaces. Using numerical methods, we detect critical behavior at the threshold of singularity formation resembling the one of gravitational collapse. In particular, the mean curvature simulation of a one-parameter family of initial data reveals the exi…

Study shows convergence of cscK surfaces in Hilbert scheme.

For any elliptic K3 surface $\mathfrak{F}: \mathcal{K} \rightarrow \mathbb{P}^1$, we construct a family of collapsing Ricci-flat Kähler metrics such that curvatures are uniformly bounded away from singular fibers, and which Gromov-Hausdorff limit to $\mathbb{P}^1$ equipped with the McLean metric. There are well-known e…

Study curve shortening flow on Riemann surfaces with conical singularities.

This note is a summary of our work [OO] which provides an explicit and global moduli-theoretic framework for the collapsing of Ricci-flat Kahler metrics and we use it to study especially the K3 surfaces case. For instance, it allows us to discuss their Gromov-Hausdorff limits along any sequences, which are even not nec…

Constructs special Lagrangian submanifolds in Calabi-Yau 3-folds.

We construct large families of new collapsing hyperkähler metrics on the K3 surface. The limit space is the quotient of a flat 3-torus by an involution. Away from finitely many exceptional points the collapse occurs with bounded curvature. There are at most 24 exceptional points where the curvature concentrates, which …

New collapsing mechanism for G2-manifolds discovered.

I study Gromov-Hausdorff limits of complex curves endowed with singular flat metrics of constant diameter. I formulate a criterion that the limit is collapsed in terms of a certain piecewise affine weight function on the dual intersection complex of a semi-stable model of the degeneration introduced by Kontsevich and S…

Ricci flow singularities on compact Kähler surfaces are of Type I.

New examples show strong Kato limits can be branching and not satisfy known conditions.

Gravitational instantons collapse to a punctured plane with a special Kahler metric.

We investigate the Chern-Ricci flow, an evolution equation of Hermitian metrics generalizing the Kahler-Ricci flow, on elliptic bundles over a Riemann surface of genus greater than one. We show that, starting at any Gauduchon metric, the flow collapses the elliptic fibers and the metrics converge to the pullback of a K…

Uniform estimates prove convergence of Chern-Ricci flow on complex surfaces.

Study heat flow on collapsing K3 surfaces, handling conic singularities.

Einstein metrics are blocked by manifold features and group growth.

The Degasperis-Procesi equation's solutions define pseudospherical metrics and can lead to surface collapse.

We investigate the Chern-Ricci flow, an evolution equation of Hermitian metrics, on Inoue surfaces. These are non-Kahler compact complex surfaces of type Class VII. We show that, after an initial conformal change, the flow always collapses the Inoue surface to a circle at infinite time, in the sense of Gromov-Hausdorff…

We analyze the topology and geometry of a polyhedron of dimension 2 according to the minimum size of a cover by PL collapsible polyhedra. We provide partial characterizations of the polyhedra of dimension 2 that can be decomposed as the union of two PL collapsible subpolyhedra in terms of their simple homotopy type and…

Uniform diameter bounds for Calabi-Yau fibrations with singular fibers.

We propose a geometric inequality for two-dimensional spacelike surfaces in the Schwarzschild spacetime. This inequality implies the Penrose inequality for collapsing dust shells in general relativity, as proposed by Penrose and Gibbons. We prove that the inequality holds in several important cases.

Let $X\rightarrow {\mathbb P}^1$ be an elliptically fibered $K3$ surface, admitting a sequence $ω_{i}$ of Ricci-flat metrics collapsing the fibers. Let $V$ be a holomorphic $SU(n)$ bundle over $X$, stable with respect to $ω_i$. Given the corresponding sequence $Ξ_i$ of Hermitian-Yang-Mills connections on $V$, we prove …

Constructs initial data for multiple black holes with specified ADM parameters.

The paper proposes a new framework to generate synthetic data with human-like imperfections to prevent model collapse.

Solves capillary $L_p$-Christoffel-Minkowski problem in half-space.

Paper proves existence of anisotropic dynamical horizons in gravitational collapse.

We investigate the behaviour of vertices and inflexions on 1-parameter families of curves on smooth surfaces in the 3-space, which include a singular member. In particular, we discuss the context where the curves evolve as sections of a smooth surface by parallel planes. More precisely we will trace the patterns of inf…

We solve Einstein vacuum equations in a spacetime region up to the "center" of gravitational collapse. Within this region, we construct a sequence of marginally outer trapped surfaces (MOTS) with areas going to zero. These MOTS form a marginally outer trapped tube (apparent horizon). It emerges from a point and is smoo…

Stability of Morse index for harmonic maps on degenerating surfaces analyzed.

We discuss an alternative approach to the uniformisation problem on surfaces with boundary by representing conformal structures on surfaces $M$ of general type by hyperbolic metrics with boundary curves of constant positive geodesic curvature. In contrast to existing approaches to this problem, the boundary curves of o…

Study quantizes topological numbers on degenerating Einstein manifolds.

We improve Gross-Wilson's local estimates to global ones. As an application, we study the blow-up limits of the degenerating Calabi-Yau metrics on singular fibers.

Formula proves Euler characteristic of singularized surfaces.

Paper constructs non-symmetric collapsing spacetimes without symmetries.