arXiv research

Hyperkahler manifolds with round Kahler cones have unique bimeromorphic models.

Inverse curvature flows shape star-shaped hypersurfaces into spheres.

The study identifies surfaces with Maslovian normal bundles.

We study quiver gauge theories on the round and squashed seven-spheres, and orbifolds thereof. They arise by imposing $G$-equivariance on the homogeneous space $G/H=\mathrm{SU}(4)/\mathrm{SU}(3)$ endowed with its Sasaki-Einstein structure, and $G/H=\mathrm{Sp}(2)/\mathrm{Sp}(1)$ as a 3-Sasakian manifold. In both cases …

Given a convex cone in the \emph{prescribed} warped product, we consider hypersurfaces with boundary which are star-shaped with respect to the center of the cone and which meet the cone perpendicularly. If those hypersurfaces inside the cone evolve along the inverse mean curvature flow, then, by using the convexity of …

We consider strictly convex hypersurfaces with the boundary which meets a strictly convex cone perpendicularly. We prove that if these hypersurfaces expand inside this cone, driven by the power of the Gauss curvature, then the evolution exists for all the time and the evolving hypersurfaces converge smoothly to a piece…

We consider a smooth Euclidean solid cone endowed with a smooth homogeneous density function used to weight Euclidean volume and hypersurface area. By assuming convexity of the cone and a curvature-dimension condition we prove that the unique compact, orientable, second order minima of the weighted area under variation…

The study of Einstein manifolds with curvature operator cone conditions.

Paper refines Einstein manifold result with cone curvature condition.

The study finds conditions for area-minimizing cones over submanifolds.

We show that each end of a noncompact self-shrinker in $\mathbb{R}^3$ of finite topology is smoothly asymptotic to either a regular cone or a self-shrinking round cylinder.

Minimal Lagrangian diffeomorphisms between spherical surfaces are rigid.

Study on sphere immersions and their stability indices.

The paper studies how surfaces evolve in a cone under a specific flow.

Study submanifolds in null hypersurfaces of generalized Robertson-Walker spacetimes.

For a surface in 3-sphere, by identifying the conformal round 3-sphere as the projectivized positive light cone in Minkowski 5-spacetime, we use the conformal Gauss map and the conformal transform to construct the associate homogeneous 4-surface in Minkowski 5-spacetime. We then derive the local fundamental theorem for…

Study on stability of surfaces in null cones under area-preserving variations.

New expanding Ricci solitons found starting in dimension four.

We classify simply connected compact Sasaki manifolds of dimension $2n+1$ with positive transverse bisectional curvature. In particular, the Kähler cone corresponding to such manifolds must be bi-holomorphic to $\C^{n+1}\backslash \{0\}$. As an application we recover the Mori-Siu-Yau theorem on the Frankel conjecture a…

New solutions found for a complex boundary problem.

Proves uniqueness of blowups for forced mean curvature flow.

For any asymptotically conical self-shrinker with entropy less than or equal to that of a cylinder we show that the link of the asymptotic cone must separate the unit sphere into exactly two connected components, both diffeomorphic to the self-shrinker. Combining this with recent work of Brendle, we conclude that the r…

We investigate complete minimal submanifolds $f\colon M^3\to\Hy^n$ in hyperbolic space with index of relative nullity at least one at any point. The case when the ambient space is either the Euclidean space or the round sphere was already studied in \cite{dksv1} and \cite{dksv2}, respectively. If the scalar curvature i…

The paper mainly concerns the structure at infinity for complete gradient shrinking Ricci solitons. It is shown that for such a soliton with bounded curvature, if the round cylinder $\mathbb{R}\times \mathbb{S}^{n-1}/Γ$ occurs as a limit for a sequence of points going to infinity along an end, then the end is asymptoti…

One way to generalize the boundary Yamabe problem posed by Escobar is to ask if a given metric on a compact manifold with boundary can be conformally deformed to have vanishing $σ_k$-curvature in the interior and constant $H_k$-curvature on the boundary. When restricting to the closure of the positive $k$-cone, this is…

Study on holomorphic curves in 6-sphere with boundary conditions.

Gradient descent with biased rounding errors converges faster under certain conditions.

Introduces holed cone structures to generalize cone structures on 3-manifolds.

Unique cylindrical tangent cone for Simons' hypersurface found.

Round surgery diagrams represent 3-manifolds in $S^3$.

This is a continuation of the previous articles on Kahler cone metrics. In this article, we introduce weighted function spaces and provide a self-contained treatment on cone angles in the whole interval $(0,1]$. We first construct geodesics in the space of Kahler cone metrics (cone geodesics). We next determine the ver…

Given a closed orientable Euclidean cone 3-manifold C with cone angles less than or equal to pi, and which is not almost product, we describe the space of constant curvature cone structures on C with cone angles less than pi. We establish a regeneration result for such Euclidean cone manifolds into spherical or hyperbo…

Study strict stability of cones with isolated singularities.

For 3-dimensional hyperbolic cone structures with cone angles $θ$, local rigidity is known for $0 \leq θ\leq 2π$, but global rigidity is known only for $0 \leq θ\leq π$. The proof of the global rigidity by Kojima is based on the fact that hyperbolic cone structures with cone angles at most $π$ do not degenerate in defo…

Weiss and, independently, Mazzeo and Montcouquiol recently proved that a 3--dimensional hyperbolic cone-manifold (possibly with vertices) with all cone angles less than $2π$ is infinitesimally rigid. On the other hand, Casson provided 1998 an example of an infinitesimally flexible cone-manifold with some of the cone an…

Contact round surgeries on $(\mathbb{S}^3,ξ_{st})$ help in constructing and understanding contact 3-manifolds.

In this article, we extend Huisken's theorem that convex surfaces flow to round points by mean curvature flow. We construct certain classes of mean convex and non-mean convex hypersurfaces that shrink to round points and use these constructions to create pathological examples of flows. We find a sequence of flows that …

The study finds billiard trajectories with infinitely many reflections in certain cones.

Optimizes sample and round complexity in adaptive sampling from multiple distributions.

The SCMU algorithm computes cone factorizations for symmetric cones, improving upon existing methods.

We describe some properties of noncompact Euclidean cone manifolds with cone angles less than c less than 2pi and singular locus a submanifold. More precisely, we describe its structure outside a compact set. As a corollary we classify those with cone angles less than 3pi/2 and those with all cone angles equal to 3pi/2…

We prove that every closed oriented 3-manifold admits a hyperbolic cone-manifold structure with cone-angle arbitrarily close to 2pi.

Consider an analytic map of a neighborhood of 0 in a vector space to a Euclidean space. Suppose that this map takes all germs of lines passing through 0 to germs of circles. Such a map is called rounding. We introduce a natural equivalence relation on roundings and prove that any rounding, whose differential at 0 has r…

New findings show infinitely many knots cannot be smoothly round handle slices.

We discuss the integrability of orthogonal almost complex structures on Riemannian products of even-dimensional round spheres and give a partial answer to the question raised by E. Calabi concerning the existence of complex structures on a product manifold of a round 2-sphere and a round 4-sphere.

This work investigates how multi-round reasoning improves LLM performance.

New Calabi-Yau metrics with conical singularities are created near complex lines.

Optimal family of Calabi-Yau cone metrics found for toric Kähler cones.