Generalized Kazdan-Warner equations associated with a linear action of a torus on a complex vector space
Natsuo Miyatake
备注：
The numbering of theorems, propositions, examples, corollaries, lemmas, definitions, and remarks has been made the same as in the published version. 23 pages
We introduce generalized Kazdan-Warner equations on Riemannian manifolds associated with a linear action of a torus on a complex vector space. We show the existence and the uniqueness of the solution of the equation on any compact Riemannian manifold. As an application, we give a new proof of a theorem of Baraglia which asserts that a cyclic Higgs bundle gives a solution of the periodic Toda equation.
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.
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. We relate our results to the work of Hong, Jost, and Struwe on classical vortices, and that of Haydys and Walpuski on the Seiberg-Witten equations with multiple spinors.
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 flat, $v^{(2k)}=(-2)^{-k}σ_k$, $G_{2r}(g)=\frac{4^r(n-r)!r!}{(n-2r)!}σ_r$, and our results reduce to earlier ones established by Viaclovsky and by the second author.
In this paper, we study the sign-changing Kazdan-Warner's problem on two dimensional closed Riemannian manifold with negative Euler number $χ(M)<0$. We show that once, the direct method on convex sets is used to find a minimizer of the corresponding functional, then there is another solution via a use of the variational method of mountain pass. In conclusion, we show that there are at least two solutions to the Kazdan-Warner's problem on two dimensional Kazdan-Warner equation provided the prescribed function changes signs and with this average negative.
We study Kazdan-Warner equations on a connected finite graph via the method of the degree theory. Firstly, we prove that all solutions to the Kazdan-Warner equation with nonzero prescribed function are uniformly bounded and the Brouwer degree is well defined. Secondly, we compute the Brouwer degree case by case. As consequences, we give new proofs of some known existence results for the Kazdan-Warner equation on a connected finite graph.
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 of $h$, is negative. Conversely, if $\overline{h}<0$, then there exists a number $c_-(h)<0$, such that the Kazdan-Warner equation is solvable for every $0>c>c_-(h)$ and it is not solvable for $c<c_-(h)$. Moreover, if $h\leq0$ and $h\not\equiv0$, then $c_-(h)=-\infty$. Inspired by…
In this article, we first show that given a smooth function $ S $ either on closed manifolds $ (M, g) $ or compact manifolds $ (\bar{M}, g) $ with non-empty boundary, both for dimensions at least $ 3 $, the condition $ S \equiv 0 $, or $ S $ changes sign and $ \int_{M} S \dvol < 0 $ (with zero mean curvature if the boundary is not empty), is both the necessary and sufficient condition for prescribing scalar curvature problems within conformal class $ [g] $, provided that the first eigenvalue of the conformal Laplacian is zero. We then extend the same necessary and sufficient condition, in term…
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 it implies that there exists a unique solution to the equation $Lu=f$ for any given $f\in C(V)$. We also prove that the equation $Δ_pu=\overline{f}-f$ has a solution which is unique up to a constant, where $\overline{f}$ is the average of $f$. With the help of these results, we fi…
We consider an evolution problem associated to the Kazdan-Warner equation on a closed Riemann surface $(Σ,g)$ \begin{align*} -Δ_{g}u=8π\left(\frac{he^{u}}{\int_Σhe^{u}{\rm d}μ_{g}}-\frac{1}{\int_Σ{\rm d}μ_{g}}\right) \end{align*} where the prescribed function $h\geq0$ and $\max_Σh>0$. We prove the global existence and convergence under additional assumptions such as \begin{align*} Δ_{g}\ln h(p_0)+8π-2K(p_0)>0 \end{align*} for any maximum point $p_0$ of the sum of $2\ln h$ and the regular part of the Green function, where $K$ is the Gaussian curvature of $Σ$. In particular, this gives a new pro…
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 captured and extended by a single identity, due to Schoen in 1988. In each of the three cases the identity requires and involves an infinitesimal conformal symmetry. For structures with such a conformal vector field, we develop a very wide, and essentially complete, extension of this p…
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）；Departamento de Matemática – Universidade Federal da Paraíba（帕拉伊巴联邦大学数学系）；Instituto de Ciência e Tecnologia – Unifesp（圣保罗联邦大学科学与技术学院）
Let $(M,g)$ be a compact Riemann surface with unit area, $h\in C^{\infty}(M)$ a function which is positive somewhere, $ρ>0$, $p_i\in M$ and $α_i\in(-1,+\infty)$ for $i=1,\cdots,\ell$, we consider the mean field equation \begin{align*} Δv + 4π\sum_{i=1}^{\ell}α_i\left(1-δ_{p_i}\right) = ρ\left(1-\frac{he^v}{\int_Mhe^vdμ}\right), \end{align*} on $M$, where $Δ$ and $dμ$ are the Laplace-Beltrami operator and the area element of $(M,g)$ respectively. Using variational method and blowup analysis, we prove some existence results in the critical case $ρ=8π(1+\min\{0,\min_{1\leq i\leq\ell}α_i\})$. Thes…
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 solutions to the symplectic vortex equations and effective divisors.
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 Navier-Stokes equations on manifolds and Kazdan-Warner-Bourguignon-Ezin identity for conformal Killing vector fields. We also provide two new criterions on the existence of Killing vector fields. The similar flow to finding holomorphic vector fields on Kähler manifolds will be st…
In this article, we first show that for all compact Riemannian manifolds with non-empty smooth boundary and dimension at least 3, there exists a metric, pointwise conformal to the original metric, with constant scalar curvature in the interior, and constant scalar curvature on the boundary by considering the boundary as a manifold of its own with dimension at least 2. We then show a series of prescribed scalar curvature results in the interior and on the boundary, with pointwise conformal deformation. These type of results is both an analogy and an extension of Kazdan and Warner's "Trichotomy …
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 $\int_M f dμ_{{\it g}_0} <0$, then there exists at least one solution $u$ to the prescribed $Q$-curvature equation \[ \mathbf{P}_{g_0} u = f e^{4u}, \] where $\mathbf{P}_{g_0}$ is the Paneitz operator which is positive with kernel consisting of constant functions. In this paper, 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 the famous problem of prescribing scalar curvature on $§^n$ for conformal metrics, posed by Nirenberg and Kazdan-Warner. This construction lets us translate into the hyperbolic setting the known results for the scalar curvature problem, and also provides a hypersurface theory inte…
