arXiv research

Study asymptotic behavior of Weingarten surfaces at infinity.

Study of Gordian graphs' behavior at infinity for various local moves.

Study exterior maximal surfaces with precise asymptotic behavior.

The paper explores integral formulas and behavior of spherical functions.

Study precise asymptotic behavior of functions in singular metric spaces.

Study geodesics in conformally compact manifolds, showing smoothness and asymptotic behavior.

Study shows quantum behavior near infinity in metric asymptotics.

New example of non-Kähler soliton with Kähler-like behavior at infinity.

In this paper, we study the limiting behavior of the Brown-York mass and Hawking mass along nearly round surfaces at infinity of an asymptotically flat manifold. Nearly round surfaces can be defined in an intrinsic way. Our results show that the ADM mass of an asymptotically flat 3-manifold can be approximated by some …

Study on harmonic functions in spaces with collapsing behaviors.

We investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in $M\times \mathbb R_+$, where $M$ is a complete noncompact Riemannian manifold. Under specific assumptions, we establish existence of solutions satisfying prescribed conditions at infinity, depending on the…

The paper explores when end summing is unique for manifolds at infinity.

Let X be a quasiprojective manifold given by the complement of a divisor $\bD$ with normal crossings in a smooth projective manifold $\bX$. Using a natural compactification of $X$ by a manifold with corners $\tX$, we describe the full asymptotic behavior at infinity of certain complete Kahler metrics of finite volume o…

Study shows global oscillatory solutions for Yang-Mills heat flow in 4D space.

We show precompactness results for solutions to parabolic fourth order geometric evolution equations. As part of the proof we obtain smoothing estimates for these flows in the presence of a curvature bound, an improvement on prior results which also require a Sobolev constant bound. As consequences of these results we …

Study of ends of complete gradient Schouten solitons, showing finitely many ends for shrinking and connected infinity for expanding ones.

We study constant mean curvature 1/2 surfaces in H2xR that admit a compactification of the mean curvature operator. We show that a particular family of complete entire graphs over H2 admits a structure of infinite dimensional manifold with local control on the behaviors at infinity. These graphs also appear to have a h…

Classifies ancient and expanding Ricci flows with specific groups.

We partially resolve a conjecture of Meeks on the asymptotic behavior of minimal surfaces in $\mathbb{R}^3$ with quadratic area growth.

We mainly study 3-dimensional complete gradient Ricci solitons with positive sectional curvature, whose scalar curvature attains its maximum at some point. In section 2, we estimate the area growth of level sets and the volume growth of sublevel sets of a Ricci potential. In section 3, we show that the scalar curvature…

Proves existence of Poisson equation on certain non-compact manifolds.

Study of mean curvature flows on graphs in warped product manifolds, focusing on behavior at infinity.

Study inverse mean curvature flow on entire graphs, proving finite time existence for certain asymptotic cases.

Transforms instantons to monopoles for torus products.

The paper studies curve shortening flows on non-convex surfaces.

We study the asymptotic volume ratio of non-steady gradient Ricci solitons. Moreover, a local estimate of the volume ratio is obtained for expanding solitons which satisfy $\lim_{dist(O,x)\rightarrow\infty} |Sect|\cdot dist(O,x)^2=0$. Therefore, for such a soliton, we can show that it must have $\mathbb{R}^n$ as one of…

For a closed real algebraic plane affine curve dividing its complexification and equipped with a complex orientation, the Whitney number is expressed in terms of behavior of its complexification at infinity.

The study examines the behavior of maps near the boundary of surfaces.

In this paper, we study the convexity, interior gradient estimate, Liouville type theorem and asymptotic behavior at infinity of translating solutions to mean curvature flow as well as the nonlinear flow by powers of the mean curvature.

Study proves behaviors of CMC surfaces near future null-infinity in Schwarzschild spacetime.

Study confirms asymptotic behavior of logarithmic balanced metric near infinity.

New Calabi-Yau metrics constructed with detailed geometry at infinity.

Study on eigenvalue rate of geodesic balls in asymptotically hyperbolic Einstein manifolds.

We study the blowup behavior at infinity of the normalized Kahler-Ricci flow on a Fano manifold which does not admit Kahler-Einstein metrics. We prove an estimate for the Kahler potential away from a multiplier ideal subscheme, which implies that the volume forms along the flow converge to zero locally uniformly away f…

This paper is devoted to study of transformations on metric spaces. It is done in an effort to produce qualitative version of quasi-isometries which takes into account the asymptotic behavior of the Gromov product in hyperbolic spaces. We characterize a quotient semigroup of such transformations on Teichmüller space by…

We discuss the Ricci-flat `model metrics' on $\mathbb{C}^2$ with cone singularities along the conic $\{zw=1\}$ constructed by Donaldson using the Gibbons-Hawking ansatz over wedges in $\mathbb{R}^3$. In particular we describe their asymptotic behavior at infinity and compute their energies.

We prove that the space of complete, finite volume, pinched negatively curved Riemannian metrics on a smooth high-dimensional manifold is either empty or it is highly non-connected, provided their behavior at infinity is similar.

Study finds solutions to inequality decay to zero on warped cylinders.

Study shows how weak inverse anisotropic mean curvature flow behaves at infinity.

Motivated by the equation satisfied by the extremals of certain Hardy-Sobolev type inequalities, we show sharp $L^q$ regularity for finite energy solutions of p-laplace equations involving critical exponents and possible singularity on a sub-space of $\mathbb{R}^n$, which imply asymptotic behavior of the solutions at i…

The paper is devoted to the large scale geometry of the Heisenberg group $\mathbb H$ equipped with left-invariant Riemannian distances. We prove that two such distances have bounded difference if and only if they are asymptotic, i.e., their ratio goes to one, at infinity. Moreover, we show that for every left-invariant…

We consider the asymptotic behavior of properly embedded minimal surfaces in the product of the hyperbolic plane with the line, taking into account the fact that there is more than one natural compactification of this space. This provides a better setting in which to consider the general problem of determining which cu…

In this paper we obtain the asymptotic behavior of solutions of the Klein-Gordon equation on Lorentzian manifolds $(X^\circ,g)$ which are de Sitter-like at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces. Under global assumptions on…

Study on scalar curvature decay in four-dimensional steady solitons.

Let (M,d) be a metric space. For 0<r<R, and p in M let G(p,r,R) be the group obtained by considering all loops based at p whose image is contained in the closed ball of radius r and identifying two loops if there is a homotopy betweeen them that is contained in the open ball of radius R. In this paper we study the asym…

In this paper we are concerned with a class of elliptic differential inequalities with a potential both on $\erre^m$ and on Riemannian manifolds. In particular, we investigate the effect of the geometry of the underlying manifold and of the behavior of the potential at infinity on nonexistence of nonnegative solutions.

Kahler-Ricci flow long-time behavior and initial data

Study on steady solitons' curvature behavior near infinity.