arXiv research

The sinh-Gordon equation is solved on finite, symmetric graphs.

Study on prescribing curvature on specific manifolds with negative Gauduchon degree.

We prove that, for a generic set of smooth prescription functions $h$ on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature $h$. The solution is either an embedded minimal hypersurface with integer multiplicity, or a non-minimal almost embedded hypersur…

We study the problem of existence of surfaces in ${\bf R}^3$ parametrized on the sphere ${\mathbb S}^2$ with prescribed mean curvature $H$ in the perturbative case, i.e. for $H=H_0+εH_1$, where $H_0$ is a nonzero constant, $H_1$ is a $C^2$ function and $ε$ is a small perturbation parameter.

We consider maps between Riemannian manifolds in which the map is a stationary point of the nonlinear Hodge energy. The variational equations of this functional form a quasilinear, nondiagonal, nonuniformly elliptic system which models certain kinds of compressible flow. Conditions are found under which singular sets o…

Sharp bounds and rigidity theorems for eigenvalues on manifolds.

Paper proves translating solutions for a specific flow in a product manifold.

In this paper we complete the topological description of the space of representations of the fundamental group of a punctured surface in SL(2,R) with prescribed behavior at the punctures and nonzero Euler number, following the strategy employed by Hitchin in the unpunctured case and exploiting Hitchin-Simpson correspon…

Study Kazdan-Warner equations on graphs using Brouwer degree theory.

Sharp threshold found for metric uniqueness in Riemannian Calderón-type problems.

The paper solves conditions for prescribing scalar and Gauss curvatures on manifolds with zero first eigenvalue.

In this paper, we develop a min-max theory for the construction of constant mean curvature (CMC) hypersurfaces of prescribed mean curvature in an arbitrary closed manifold. As a corollary, we prove the existence of a nontrivial, smooth, closed, almost embedded, CMC hypersurface of any given mean curvature $c$. Moreover…

Study axisymmetric $σ_k$-Nirenberg problem on spheres.

We study analysis aspects of the sixth order GJMS operator $P_g^6$. Under conformal normal coordinates around a point, the expansions of Green's function of $P_g^6$ with pole at this point are presented. As a starting point of the study of $P_g^6$, we manage to give some existence results of prescribed $Q$-curvature pr…

The nonzero level sets of a homogeneous, logarithmically homogeneous, or translationally homogeneous function are affine spheres if and only if the Hessian determinant of the function is a multiple of a power or an exponential of the function. In particular, the nonzero level sets of a homogeneous polynomial are proper…

Study stabilizes second-order systems to first-order dynamics.

This paper improves lower bounds for the first nonzero Steklov eigenvalue on compact manifolds.

Let $G$ be a compact connected Lie group and $H$ a closed subgroup of $G$. Suppose the homogeneous space $G/H$ is effective and has dimension 3 or higher. Consider a $G$-invariant, symmetric, positive-semidefinite, nonzero (0,2)-tensor field $T$ on $G/H$. Assume that $H$ is a maximal connected Lie subgroup of $G$. We p…

In this paper, we employ a nonlocal $Q$-curvature flow inspired by Gursky-Malchiodi's work \cite{gur_mal} to solve the prescribed $Q$-curvature problem on a class of closed manifolds: For $n \geq 5$, let $(M^n,g_0)$ be a smooth closed manifold, which is not conformally diffeomorphic to the standard sphere, satisfying e…

In this paper, we show that the peeling property still holds for Bondi-Sachs metrics with nonzero cosmological constant under the boundary condition given by Sommerfeld's radiation condition together with three nontrivial $Λ$-independent functions $B$, $a$, $b$. This should indicate the new boundary condition is natura…

Proves optimal regularity for sphere minimizers in 3-sphere.

Study on half spheres solves Nirenberg problem with complex blow-up analysis.

Proves existence and uniqueness of Killing graphs with prescribed curvature.

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

We use a phase space analysis to give some classification results for rotational hypersurfaces in $\mathbb{R}^{n+1}$ whose mean curvature is given as a prescribed function of its Gauss map. For the case where the prescribed function is an even function in $\mathbb{S}^n$, we show that a Delaunay-type classification hold…

Classification of surfaces with prescribed mean curvature in Heisenberg space and SL2(R).

In this paper, we consider the problem of prescribing scalar curvature on n-sphere. Assume that the candidate curvature function $f$, which is allowed to change sign, satisfies some kind of Morse index or symmetry condition. By studying the well-known scalar curvature flow, we are able to prove that the flow converges …

Develops variational approach for Kähler-Einstein metrics with prescribed singularities on Fano manifolds.

Study local curvature estimates and existence of conformal metrics on noncompact manifolds.

We prove existence of compact spacelike hypersurfaces with prescribed k - curvature in de Sitter space, where the prescription function depends on both space and the tilt function.

We relate the jumps of the signature function of a link to the roots of its first nonzero higher Alexander polynomial.

Researchers find examples of real Bott manifolds with nonzero dual Stiefel-Whitney class wbar_{n-ahat(n)} for all n nonzero mod 4.

In this work, we study the Yamabe flow corresponding to the prescribed scalar curvature problem on compact Riemannian manifolds with negative scalar curvature. The long time existence and convergence of the flow are proved under appropriate conditions on the prescribed scalar curvature function.

Conditions for scalar curvature on compact manifolds under conformal deformation.

Study reconstructs Morse-Bott functions with specific preimage conditions on 3D manifolds.

Theory proves existence of hypersurfaces with prescribed curvature.

In this paper we construct complete simply connected minimal surfaces with a prescribed coordinate function. Moreover, we prove that these surfaces are dense in the space of all minimal surfaces with this coordinate function (with the topology of the smooth convergence on compact sets).

Study finds loops with specific curvature exist using Hardy's inequality.

In this paper, we prove the existence of hypersurfaces in the Euclidean space with prescribed boundary and whose k-th Weingarten curvature equals a given function that depends on the normal of the hypersurface. The proof is based on the solvability of a fully nonlinear elliptic PDE. The required a priori estimates are …

Estimates for plate eigenvalues with nonzero Poisson's ratio.

Global existence and convergence proved for Kazdan-Warner equation with non-negative prescribed function.

We study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Ricci tensor. We prove an existence theorem for a wide class of symmetric functions on manifolds with positive Ricci curvature, provided the conformal class admits an admissible metric.

Solves modified Schouten tensor problems in conformal metric classes.

The paper shows that random frames have full spark with high probability.

Study rotational surfaces with prescribed Gauss curvature in 3D space.

Motivated by a simple model for elastic cell membranes, we minimize the Willmore functional among two-dimensional spheres embedded in R^3 with prescribed isoperimetric ratio.

The nonzero level sets in $n$-dimensional flat affine space of a translationally homogeneous function are improper affine spheres if and only if the Hessian determinant of the function is equal to a nonzero constant multiple of the $n$th power of the function. The exponentials of the characteristic polynomials of certa…

We prove the existence of metrics with prescribed $Q$-curvature under natural assumptions on the sign of the prescribing function and the background metric. In the dimension four case, we also obtain existence results for curvature forms requiring only restrictions on the Euler characteristic. Moreover, we derive a pre…