arXiv research

Characterizes rotational solitons for curve shortening flow on revolution surfaces.

The paper classifies expanding gradient Yamabe solitons based on scalar curvature.

The study proves rotationally symmetric property of certain shrinking gradient Yamabe solitons.

New examples of solitons found using submersion techniques.

The paper classifies solitons in a curved product space.

The paper classifies hypersurfaces in product spaces with specific curvature properties and finds that only rotational ones admit almost Ricci solitons.

We show that expanding Kähler-Ricci solitons which have positive holomorphic bisectional curvature and are asymptotic to Kähler cones at infinity must be the U(n)-rotationally symmetric expanding solitons constructed by Cao.

New gravitational solitons and infinite topological manifolds found.

We investigate self-similar solutions to the inverse mean curvature flow in Euclidean space. In the case of one dimensional planar solitons, we explicitly classify all homothetic solitons and translators. Generalizing Andrews' theorem that circles are the only compact homothetic planar solitons, we apply the Hsiung-Min…

Researchers classify translators and rotators in hyperbolic 3-space for mean curvature flow.

We provide a detailed description of solutions of Curve Shortening in $\R^n$ that are invariant under some one-parameter symmetry group of the equation, paying particular attention to geometric properties of the curves, and the asymptotic properties of their ends. We find generalized helices, and a connection with curv…

The paper classifies invariant gradient $k$-Yamabe solitons in pseudo-Euclidean spaces.

Classifies timelike translating solitons in Minkowski space.

Let (M,g) be a steady gradient Ricci soliton of dimension n \geq 4 which has positive sectional curvature and is asymptotically cylindrical. Under these assumptions, we show that (M,g) is rotationally symmetric. In particular, our result applies to steady gradient Ricci solitons in dimension 4 which are κ-noncollapsed …

Classifies and constructs translators for curvature flows.

This paper has been withdrawn by the author due to an error

Ancient solutions to curve shortening flow are constructed and analyzed.

Let (M,g) be a three-dimensional steady gradient Ricci soliton which is non-flat and κ-noncollapsed. We prove that (M,g) is isometric to the Bryant soliton up to scaling. This solves a problem mentioned in Perelman's first paper.

In this paper we show that all conformal metrics to a pseudo-euclidean space invariant under the translation group, and all the conformal metrics product manifold also invariant by translation where F m it is Ricci flat semi-Riemannian manifold, are gradient Ricci almost soliton. We also proved that all conformal metri…

The study finds complete translating solitons for certain powers of Gaussian curvature in Riemannian products.

This paper classifies Kähler manifolds with specific Einstein-type properties.

In a recent paper, Brendle showed the uniqueness of the Bryant soliton among 3-dimensional $κ$-solutions. In this paper, we present an alternative proof for this fact and show that compact $κ$-solutions are rotational symmetric. Our proof arose from independent work relating to our Strong Stability Theorem for singular…

New proof shows symmetry for certain curved surfaces in higher dimensions.

The study examines gradient almost Yamabe solitons in warped product manifolds and their geometric properties.

A $λ$-translating soliton with density vector $\vec{v}$ is a surface in Euclidean space whose mean curvature $H$ satisfies $2H=2λ+\langle N,\vec{v}\rangle$, where $N$ is the Gauss map. We classify all $λ$-translating solitons that are invariant by a one-parameter group of translations and a one-parameter group of rotat…

For a polygon in Euclidean space we consider a transformation T which is obtained by applying the midpoints polygon construction twice and using an index shift. For a closed polygon this is a curve shortening process. A polygon is called (affine) soliton of the transformation T if its image under T is an affine image o…

New steady Kähler-Ricci solitons found on orbifolds.

The paper proves nonexistence results for translating solitons in r-mean curvature flow.

The special nature of gradient Yamabe soliton equation which was first observed by Cao-Sun-Zhang\cite{CaoSunZhang} shows that a complete gradient Yamabe soliton with non-constant potential function is either defined on the Euclidean space with rotational symmetry, or on the warped product of the real line with a manifo…

The paper explores non-minimal solitons in the sphere with unique properties.

The paper constructs surfaces with prescribed mean curvature in a specific space.

Ancient solutions to Ricci flow in higher dimensions are mostly cylinders or solitons.

Study on mixed Killing vector fields on Cigar Ricci-Bourguignon solitons.

New families of translating solitons found in hyperbolic space.

Let $(M^n,g,\nabla f)$, $n\geq 3$, be an expanding gradient Ricci soliton with nonnegative sectional curvature whose asymptotic cone is isometric to $C(\mathbb{S}^{n-1}(c))$ where $\mathbb{S}^{n-1}(c)$ is the standard $(n-1)$-sphere of curvature $1/c^2$, with $c\in(0,1)$. We prove that if the convergence to the asympto…

The paper classifies periodic solitons in curve flows on the light-cone.

Study precise asymptotics of noncompact Type-IIb solutions to mean curvature flow.

Study invariant $λ$-translators in Lorentz-Minkowski space.

We construct embedded ancient solutions to mean curvature flow related to certain classes of unstable minimal hypersurfaces in $\mathbb{R}^{n+1}$ for $n \geq 2$. These provide examples of mean convex yet nonconvex ancient solutions that are not solitons, meaning that they do not evolve by rigid motions or homotheties. …

We study stationary configurations mimicking nonholonomic locally anisotropic black rings (for instance, with ellipsoidal polarizations and/or imbedded into solitonic backgrounds) in three/six dimensional pseudo-Finsler/ Riemannian spacetimes. In the asymptotically flat limit, for holonomic configurations, a subclass o…

Classifies ancient solutions to curvature flows, finding two main types.

In this article we investigate the dynamics of special solutions to the surface diffusion flow of idealised ribbons. This equation reduces to studying the curve diffusion flow for the profile curve of the ribbon. We provide: (1) a complete classification of stationary solutions; (2) qualitative results on shrinkers, tr…

Constructing solutions to geometric flows with rotational symmetry.

In [4], we proved that every noncompact ancient $κ$-solution to the Ricci flow in dimension $3$ is either locally isometric to a family of shrinking cylinders, or isometric to the Bryant soliton. In the same paper, we announced that the same method implies that compact ancient $κ$-solutions are rotationally symmetric. …

We prove that the Ricci flow on CP^n blown-up at one point starting with any rotationally symmetric Kahler metric must develop Type I singularities. In particular, if the total volume does not go to zero at the singular time, the parabolic blow-up limit of the Type I Ricci flow along the exceptional divisor is a comple…

Classifies regularity for Lagrangian mean curvature type equations.

In this paper, geometric characterizations of conformally flat and radially flat hypersurfaces in $\mathbb{S}^n \times \mathbb{R}$ and $\mathbb{H}^n \times \mathbb{R}$ are given by means of their extrinsic geometry. Under suitable conditions on the shape operator, we classify conformally flat hypersurfaces in terms of …

We will prove the non-existence of positive radially symmetric solution of the nonlinear elliptic equation $\frac{n-1}{m}Δv^m+αv+βx\cdot\nabla u=0$ in $R^n$ when $n\ge 3$, $0<m\le\frac{n-2}{n}$, $α<0$ and $β\le 0$. Let $n\ge 3$ and $g=v^{\frac{4}{n+2}}dx^2$ be a metric on $\R^n$ where $v$ is a radially symmetric soluti…