arXiv research

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

Extends existence results for scalar curvature on conical manifolds.

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…

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

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 …

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

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 …

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 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…

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

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

New Kazdan-Warner problem for equivariant metrics on manifolds.

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

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

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

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.…

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

The paper tackles prescribing discrete Gaussian curvature on polyhedral surfaces.

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

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 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.

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…

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…

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 …

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 …

This paper studies torsion obstructions to complex sections on manifolds.

Fewer obstructions for small graphs in knotless embedding.

3028 obstructions found for embedding without knots.

The paper develops obstructions for embedding 2D complexes into 4D space.

Global obstructions found for conformally Einstein metrics in 6D.

Complete surgery obstructions for manifolds with finite fundamental group, disproving a conjecture.

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

Given a polarized manifold there are obstructions for asymptotic Chow semistability described as integral invariants. One of them is an obstruction to the existence for the first Chern class of the polarization to admit a constant scalar curvature Kähler (cscK) metric. A natural question is whether or not the other obs…

The article classifies cubiquitous sublattices and applies them to branched covers.

For a 3-manifold with torus boundary admitting an appropriate involution, we show that Khovanov homology provides obstructions to certain exceptional Dehn fillings. For example, given a strongly invertible knot in S^3, we give obstructions to lens space surgeries, as well as obstructions to surgeries with finite fundam…

New obstructions found for smooth desingularization of compact Einstein orbifolds.

For $\ell >1$, we develop $L^{(2)}$-signature obstructions for $(4\ell-3)$-dimensional knots with metabelian knot groups to be doubly slice. For each $\ell>1$, we construct an infinite family of knots on which our obstructions are non-zero, but for which double sliceness is not obstructed by any previously known invari…

Developing a singular dimension descent method for positive scalar curvature obstructions

Study Euler obstruction of 1-forms on determinantal singularities.

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…