arXiv research

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

Solves generalized Kazdan-Warner equations on foliated manifolds.

Let $G=(V,E)$ be a connected finite graph. In this short paper, we reinvestigate the Kazdan-Warner equation $Δu=c-he^u$ with $c<0$ on $G$, where $h$ defined on $V$ is a known function. Grigor'yan, Lin and Yang \cite{GLY} showed that if the Kazdan-Warner equation has a solution, then $\overline{h}$, the average value …

New equations on manifolds linked to torus actions, proving existence and uniqueness.

Two methods find at least two solutions to Kazdan-Warner's problem on surfaces.

We study the limiting behaviour of solutions to abelian vortex equations when the volume of the underlying Riemann surface grows to infinity. We prove that the solutions converge smoothly away from finitely many points. The proof relies on a priori estimates for functions satisfying generalised Kazdan-Warner equations.…

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

Study noncompact manifolds' Chern scalar curvatures, proving existence and multiplicity.

The paper proves existence of solutions to Kazdan-Warner equations on finite graphs.

The Kazdan-Warner problem is solved for Riemann surfaces with smooth boundaries.

We obtain a Hitchin-Kobayashi-type correspondence for symplectic vortex equations, with the target a Kahler cone over a compact Sasakian manifold. We show that the correspondence reduces to studying the existence and uniqueness of Kazdan-Warner equations. Using this, we construct a map between the moduli space of solut…

Let $G=(V,E)$ be a connected finite graph and $C(V)$ be the set of functions defined on $V$. Let $Δ_p$ be the discrete $p$-Laplacian on $G$ with $p>1$ and $L=Δ_p-k$, where $k\in C(V)$ is positive everywhere. Consider the operator $L:C(V)\rightarrow C(V)$. We prove that $-L$ is one to one, onto and preserves order. So i…

Study identifies obstructions for solving a 4th-order boundary problem.

Any 2-dim Riemannian manifold with spherical topology can be embedded isometrically into a lightcone of the Minkowski spacetime. We apply this fact to give a proof of the Kazdan-Warner identity.

Study on compact Kähler surfaces for sign-changing curvatures.

This paper solves the Christoffel problem in hyperbolic space and its equivalent on spheres.

New Kazdan-Warner problem for equivariant metrics on manifolds.

Directly applies Kazdan--Warner results to prescribe scalar curvature on bundles.

The paper proves quasi-Einstein structures on manifolds admit Killing vector fields and provides new examples.

The paper tackles prescribing discrete Gaussian curvature on polyhedral surfaces.

In this paper we introduce and study a geometric heat flow to find Killing vector fields on closed Riemannian manifolds with positive sectional curvature. We study its various properties, prove the global existence of the solution of this flow, discuss its convergence and possible applications, and its relation to the …

We give a proof of the Kazdan-Warner conjecture concerning the prescribed scalar curvature problem in the null case.

In this note, we prove two Kazdan-Warner type identities involving $v^{(2k)}$, the renormalized volume coefficients of a Riemannian manifold $(M^n,g)$, and $G_{2r}$, the so-called Gauss-Bonnet curvature, and a conformal Killing vector field on $(M^n,g)$. In the case when the Riemannian manifold is locally conformally f…

We solve the modified Kazdan-Warner problem of finding metrics with prescribed scalar curvature and unit total volume.

The paper solves curvature prescription problems on balls and disks.

The classical Pohozaev identity constrains potential solutions of certain semilinear PDE boundary value problems. The Kazdan-Warner identity is a similar necessary condition important for the Nirenberg problem of conformally prescribing scalar curvature on the sphere. For dimensions $n\geq 3$ both identities are captur…

We prove a Pohozaev type identity for non-linear eigenvalue equations of the Dirac operator on Riemannian spin manifolds with boundary. As an application, we obtain that the mean curvature H of a conformal immersion S^{n}-> R^{n+1} satisfies $\int \partial_X H=0$ where X is a conformal vector field on S^{n} and where t…

We prove in this article that the local image of each conformal $Q$-curvature operator of arbitrary order on the sphere admits no scalar constraint. However, we prove that identities of Kazdan--Warner type hold for its graph.

The paper proves compactness of metrics on spheres with isolated singularities.

The paper proves a theorem for discretizing Gaussian curvature on surfaces.

Extends existence results for scalar curvature on conical manifolds.

Analog to the classical result of Kazdan-Warner for the existence of solutions to the prescribed Gaussian curvature equation on compact 2-manifolds without boundary, it is widely known that if $(M,g_0)$ is a closed 4-manifold with zero $Q$-curvature and if $f$ is any non-constant, smooth, sign-changing function with $\…

In this paper, we consider the scalar curvature of Yamabe solitons. In particular we show that, with natural conditions and non positive Ricci curvature, any complete Yamabe soliton has constant scalar curvature, namely, it is a Yamabe metric. We also show that the quadratic decay at infinity of the Ricci curvature of …

Proves existence and compactness of solutions to $σ_2$-Nirenberg problem on sphere.

In a previous paper, the authors showed that metrics which are asymptotic to Anti-de Sitter-Schwarzschild metrics with positive mass admit a unique foliation by stable spheres with constant mean curvature. In this paper we extend that result to all asymptotically hyperbolic metrics for which the trace of the mass term …

The paper solves a mean field equation on a compact Riemann surface using variational and blowup analysis.

This paper is devoted to the existence of contact forms of prescribed Webster scalar curvature on a $3-$dimensional CR compact manifold locally conformally CR equivalent to the unit sphere $\mathbb{S}^{3}$ of $\mathbb{C}^{2}$. Due to Kazdan-Warner type obstructions, conditions on the function $H$ to be realized as a We…

Our first objective in this paper is to give a natural formulation of the Christoffel problem for hypersurfaces in $H^{n+1}$, by means of the hyperbolic Gauss map and the notion of hyperbolic curvature radii for hypersurfaces. Our second objective is to provide an explicit equivalence of this Christoffel problem with t…

For a closed smooth manifold $M$ admitting a symplectic structure, we define a smooth topological invariant $Z(M)$ using almost-Kähler metrics, i.e. Riemannian metrics compatible with symplectic structures. We also introduce $Z(M, [[ω]])$ depending on symplectic deformation equivalence class $[[ω]]$. We first prove tha…

Sharp inequalities and extremals on compact Riemann surfaces with boundary.

The paper studies $\varphi$-static perfect fluid space-times in Einstein's General Relativity.

The study examines metrics with unit volume or area on manifolds with boundaries, finding critical points and solving curvature problems.

The sigma invariant is studied for torus, K3 surface, and 3d manifolds.

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

The paper studies metrics with constant scalar curvature on foliated manifolds.

The study classifies manifolds based on their geometric properties and invariants.

Study Galois groupoids of discret Painlevé equations.

Proves solvability of general inverse σ_k equations with constant coefficients.