arXiv research

We study the compact noncollapsed ancient convex solutions to Mean Curvature Flow in $\mathbb{R}^{n+1}$ with $O(1)\times O(n)$ symmetry. We show they all have unique asymptotics as $t\to -\infty$ and we give precise asymptotic description of these solutions. In particular, solutions constructed by White, and Haslhofer …

Study ancient Ricci flow solutions, proving unique asymptotic behavior.

New findings on $κ$-solutions with round cylinder as asymptotic shrinker.

Paper constructs solutions to Bogomolny equations with specific boundary and asymptotic conditions.

We show that asymptotically hyperbolic solutions of the Einstein constraint equations with constant mean curvature can be glued in such a way that their asymptotic regions are connected.

Study asymptotic behaviors of solutions near singular boundaries for the Yamabe problem.

We consider the asymptotic behaviour of positive solutions u of the conformal scalar curvature equation, Δu + n(n-2)/4 u^{(n+2)(n-2) = 0, in the neighbourhood of isolated singularities in the standard Euclidean ball. Although asymptotic radial symmetry for such solutions was proved some time ago, by Caffarelli, Gidas a…

Study on solutions near isolated singularities in 6D Yamabe equation.

We consider compact noncollapsed ancient solutions to the 3-dimensional Ricci flow that are rotationally and reflection symmetric. We prove that these solutions are either the spheres or they all have unique asymptotic behavior as $t\to-\infty$ and we give their precise asymptotic description. This description applies …

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

Static spherically symmetric solutions to the Einstein-Euler equations with prescribed central densities are known to exist, be unique and smooth for reasonable equations of state. Some criteria are also available to decide whether solutions have finite extent (stars with a vacuum exterior) or infinite extent. In the l…

This is the second part of the investigation started in [Stationary solutions and asymptotic flatness I]. We prove here that Strongly Stationary ends having cubic volume growth are Weakly Asymptotically Flat. Combined with the results of the previous paper this shows that Strongly Stationary ends are Asymptotically Fla…

We study positive solutions of the Yamabe equation with isolated singularity and prove the existence of solutions with prescribed asymptotic expansions near singular points and an arbitrarily high order of approximation.

We construct solutions with prescribed asymptotics to the Einstein constraint equations using a cut-off technique. Moreover, we give various examples of vacuum asymptotically flat manifolds whose center of mass and angular momentum are ill-defined.

Study asymptotically almost periodic solutions on real hyperbolic manifolds.

Two ancient solutions to Gauss curvature flow are identified for cylinders.

Smooth solutions found for a curvature problem in hyperbolic space.

Paper studies periodic solutions to Navier-Stokes equations on hyperbolic manifolds.

Solutions near infinity to special Lagrangian equations are asymptotic to quadratic polynomials with logarithmic terms.

In this paper we construct a parametrix for the forward fundamental solution of the wave and Klein-Gordon equations on asymptotically de Sitter spaces without caustics. We use this parametrix to obtain asymptotic expansions for solutions of the inhomogeneous equation and to obtain a uniform L^p estimate for a family of…

Study on Navier-Stokes equations on non-compact manifolds, proving existence and decay of solutions.

We construct low regularity solutions of the vacuum Einstein constraint equations. In particular, on 3-manifolds we obtain solutions with metrics in $H^s\loc$ with $s>{3\over 2}$. The theory of maximal asymptotically Euclidean solutions of the constraint equations descends completely the low regularity setting. Moreove…

Establishes scattering theory for de Sitter vacuum solutions in even dimensions.

The paper studies solutions to the Yamabe equation on asymptotically flat manifolds and their behavior at infinity.

Study on blow-up solutions for semilinear wave equations on specific manifolds.

Classifies polynomial growth solutions to drift-harmonic equations on asymptotically paraboloidal manifolds.

Paper finds invariant solutions for Plateau problem in hyperbolic space.

Because of the relevance of the results, this paper is merged into the paper titled "On the Number of Solutions to Asymptotic Plateau Problem" (arXiv:math.DG/0505593) as a new section.

By making use of the nice behavior of Hawking masses of slices of a weak solution of inverse mean curvature flow in three dimensional asymptotically hyperbolic manifolds, we are able to show that each slice of the flow is star-shaped after a long time, and then we get the regularity of the weak solution of inverse mean…

Ancient pancake solutions found for curvature flows.

$σ_k$-Yamabe equations are conformally invariant equations generalizing the classical Yamabe equation. In an earlier work YanYan Li proved that an admissible solution with an isolated singularity at $0\in \mathbb R^n$ to the $σ_k$-Yamabe equation is asymptotically radially symmetric. In this work we prove that an admis…

Study compares nodal sets of solutions to the Allen-Cahn equation.

We consider Ricci flow of complete Riemannian manifolds which have bounded non-negative curvature operator, non-zero asymptotic volume ratio and no boundary. We prove scale invariant estimates for these solutions. Using these estimates, we show that there is a limit solution, obtained by scaling down this solution at a…

We show the existence of the full compound asymptotics of solutions to the scalar wave equation on long-range non-trapping Lorentzian manifolds modeled on the radial compactification of Minkowski space. In particular, we show that there is a joint asymptotic expansion at null and timelike infinity for forward solutions…

We develop an efficient method to calibrate CDS spreads using asymptotic approximations.

Study of 4D Ricci solitons with symmetry, finding precise geometric asymptotics.

The paper confirms the existence of 5D regular static vacuum solutions with multiple black holes and Kasner asymptotics.

New exact spherically symmetric vacuum solutions found in Finsler gravity.

We construct solutions of the constraint equation with non constant mean curvature on an asymptotically hyperbolic manifold by the conformal method. Our approach consists in decreasing a certain exponent appearing in the equations, constructing solutions of these sub-critical equations and then in letting the exponent …

Study examines solutions to Jang equation on anti-de Sitter spacetimes.

Study on radial solutions of Lane-Emden system on Cartan-Hadamard manifolds.

Study of Dirichlet minimizers on manifolds with boundary and their asymptotic behavior.

New solutions found with negative mass in general relativity.

In this dissertation, we prove a number of results regarding the conformal method of finding solutions to the Einstein constraint equations. These results include necessary and sufficient conditions for the Lichnerowicz equation to have solutions, global supersolutions which guarantee solutions to the conformal constra…

In this paper we investigate the rigidity of ancient solutions of the mean curvature flow with arbitrary codimension in space forms. We first prove that under certain sharp asymptotic pointwise curvature pinching condition the ancient solution in a sphere is either a shrinking spherical cap or a totally geodesic sphere…

This note summarizes results that were obtained by the author in his habilitation thesis (arXiv:1607.08792) concerning the development of a spectral theory for simply periodic, 2-dimensional, complex-valued solutions of the sinh-Gordon equation. Spectral data for such solutions are defined for periodic Cauchy data on a…

The article models illiquid stocks using quantum calculus with asymptotic methods.

Smooth solutions and classification of Dirac-Einstein equations on 3-manifolds.