arXiv research

Ancient solutions to Kähler Ricci flow classified completely.

Complete solutions found for Toda equations on non-compact surfaces.

We study uniqueness of positive solutions to the conformal scalar curvature equation on complete Riemannian manifolds with constant negative scalar curvature. We apply the results to show that conformal transformations on certain complete Riemannian manifolds of constant negative scalar curvature are isometries. We als…

We prove the uniqueness of solutions of the Ricci flow on complete noncompact manifolds with bounded curvatures using the De Turck approach. As a consequence we obtain a correct proof of the existence of solution of the Ricci harmonic flow on complete noncompact manifolds with bounded curvatures.

Study finite curvature solutions on surfaces with nonnegative Gauss curvature.

We show that every complete entire self-shrinking solution on complex Euclidean space to the Kahler-Ricci flow must be generated from a quadratic potential.

Study on ancient Ricci flows with positive curvature, proving noncollapsedness.

In this article we study self-gravitating static solutions of the Einstein-ScalarField system in arbitrary dimensions. We discuss the existence and the non-existence of geodesically complete solutions depending on the form of the scalar field potential $V(φ)$, and provide full global geometric estimates when the soluti…

Let $(M,g)$ be a complete noncompact non-collapsing $n$-dimensional riemannian manifold, whose complex sectional curvature is bounded from below and scalar curvature is bounded from above. Then ricci flow with above as its initial data, has at most one solution in the class of complete riemannian metric with complex se…

Study shows uniqueness of solutions on complex manifolds without requiring solution decay.

This paper classifies solutions for a specific geometric problem.

Under appropriate spectral assumptions we prove two existence results for positive solutions of Lichnerowicz-type equations on complete manifolds. We also give a priori bounds and a comparison result that immediately yields uniqueness for certain classes of solutions. No curvature assumptions are involved in our analys…

Conformally compact and complete smooth solutions to the Strominger system with non vanishing flux, non-trivial instanton and non-constant dilaton using the first Pontrjagin form of the (-)-connection} on 6-dimensional non-Kaehler nilmanifold are presented. In the conformally compact case the dilaton is determined by t…

We produce solutions to the Kähler-Ricci flow emerging from complete initial metrics $g_0$ which are $C^0$ Hermitian limits of Kähler metrics. Of particular interest is when $g_0$ is Kähler with unbounded curvature. We provide such solutions for a wide class of $U(n)$-invariant Kähler metrics $g_0$ on $n$ dimensional c…

We prove the existence of a solution of the Yamabe equation on complete manifolds with finite volume and positive Yamabe invariant. In order to circumvent the standard methods on closed manifolds which heavily rely on global (compact) Sobolev embeddings we approximate the solution by eigenfunctions of certain conformal…

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

Proves planarity and convexity for ancient solutions of mean curvature flow.

Ancient solutions to Ricci flow with isotropic curvature conditions are classified.

This paper is to study the conformal scalar curvature equation on complete noncompact Riemannian manifold of nonpositive curvature. We derive some estimates and properties of supersolutions of the scalar curvature equation, and obtain some nonexistence results for complete solutions of scalar curvature equation.

In this paper we discuss the asymptotic entropy for ancient solutions to the Ricci flow. We prove a gap theorem for ancient solutions, which could be regarded as an entropy counterpart of Yokota's work. In addition, we prove that under some assumptions on one time slice of a complete ancient solution with nonnegative c…

In this paper we investigate the existence of a solution to the Poisson equation on complete manifolds with positive spectrum and Ricci curvature bounded from below. We show that if a function $f$ has decay $f=O(r^{-1-\varepsilon})$ for some $\varepsilon >0,$ where $r$ is the distance function to a fixed point, then t…

We introduce a systematic method to solve a type of Cartan's realization problem. Our method builds upon a new theory of Lie algebroids and Lie groupoids with structure group and connection. This approach allows to find local as well as complete solutions, their symmetries, and to determine the moduli spaces of local a…

We study singularity formation of complete Ricci flow solutions, motivated by two applications: (a) improving the understanding of the behavior of the essential blowup sequences of Enders-Muller-Topping on noncompact manifolds, and (b) obtaining further evidence in favor of the conjectured stability of generalized cyli…

This paper, the third in a series, completes our description of all (radial) solutions on C* of the tt*-Toda equations, using a combination of methods from p.d.e., isomonodromic deformations (Riemann-Hilbert method), and loop groups. We place these global solutions into the broader context of solutions which are smooth…

Solves Brezis' first open problem on ball solutions.

We prove an existence theorem for positive solutions to Lichnerowicz-type equations on complete manifolds with boundary and nonlinear Neumann conditions. This kind of nonlinear problems arise quite naturally in the study of solutions for the Einstein-scalar field equations of General Relativity in the framework of the …

We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as $t \to -\infty$, to two self-similar complete non-compact solutions to the Yamabe flow moving in opposite directions. They are type I ancient solutions.

Minimax solutions are weak solutions to Cauchy problems involving Hamilton--Jacobi equations, constructed from generating families quadratic at infinity of their geometric solutions. We give a complete description of minimax solutions and we classify their generic singularities of codimension not greater than 2.

We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as $t \to -\infty$, to two self-similar complete non-compact solutions to the Yamabe flow moving in opposite directions. They are type I ancient solutions.

Consider a symplectic manifold $M$, a Hamiltonian vector field $X$ and a fibration $Π:M\rightarrow N$. Related to these data we have a generalized version of the (time-independent) Hamilton-Jacobi equation: the $Π$-HJE for $X$, whose unknown is a section $σ:N\rightarrow M$ of $Π$. The standard HJE is obtained when the …

The paper studies gradient estimates for solutions of a nonlinear elliptic equation on Riemannian manifolds.

Improved Liouville theorems for ancient solutions to V-harmonic map heat flows.

Study potential theory to detect completeness of Finsler manifolds.

In this paper we provide some local and global splitting results on complete Riemannian manifolds with nonnegative Ricci curvature. We achieve the splitting through the analysis of some pointwise inequalities of Modica type which hold true for every bounded solution to a semilinear Poisson equation. More precisely, we …

We provide a complete study of existence and uniqueness of solutions to the Lichnerowicz equation in general relativity with arbitrary mean curvature.

The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton \cite{Ha1}. Later on, De Turck \cite{De} gave a simplified proof. In the later of 80's, Shi \cite{Sh1} generalized the local existence result to…

Paper shows $L^\infty$-positivity and stochastic completeness are equivalent.

We give a complete solution of a problem in submanifold theory posed and partially solved by the eminent algebraic geometer Pierre Samuel in 1947. Namely, to determine all pairs of immersions of a given manifold into Euclidean space that have the same Gauss map and induce conformal metrics on the manifold. The case of …

An affine manifold is said to be geodesically complete if all affine geodesics extend for all time. It is said to be affine Killing complete if the integral curves for any affine Killing vector field extend for all time. We use the solution space of the quasi-Einstein equation to examine these concepts in the setting o…

Paper compares solutions of Poisson equations on Riemannian manifolds with Robin boundary.

The system of weak normality equations constitutes a part in the complete system of normality equations. Solutions of each of these two systems of equations are associated with some definite classes of Newtonian dynamical systems in Riemannian manifolds. In this paper for the case of simplest flat Riemannian manifold $…

We study asymptotic behavior of positive smooth solutions of the conformal scalar curvature equation in ${\bf R}^n$. We consider the case when the scalar curvature of the conformal metric is bounded between two positive numbers outside a compact set. It is shown that the solution has slow decay if the radial change is …

The main result of this paper is: Given any constant C, there is $(ε,k,L)$ such that if a complete, orientable, noncompact odd-dimensional manifold with bounded positive sectional curvature contains a $(ε,k,L)$-neck, then the asymptotic scalar curvature ratio is bigger or equal to C. As a application we proved that the…

In this paper, we consider the Yamabe equation on a complete noncompact Riemannian manifold and find some geometric conditions on the manifold such that the Yamabe problem admits a bounded positive solution.

One of the popular approaches for low-rank tensor completion is to use the latent trace norm regularization. However, most existing works in this direction learn a sparse combination of tensors. In this work, we fill this gap by proposing a variant of the latent trace norm that helps in learning a non-sparse combinatio…

In this paper, we obtain a complete list of all self-similar solutions of inverse mean curvature flow in $\mathbb{R}^2$.

We produce longtime solutions to the Kähler-Ricci flow for complete Kähler metrics on $\Bbb C ^n$ without assuming the initial metric has bounded curvature, thus extending results in [3]. We prove the existence of a longtime bounded curvature solution emerging from any complete $U(n)$-invariant Kähler metric with non-n…

We continue the study of the operator of generalized Maxwell equations and completely discover the behavior of the solutions of the time-harmonic equations as the frequency tends to zero. Thereby, we identify degenerate operators in terms of special 'polynomially growing' solutions of a corresponding static problem, wh…