arXiv subjects
Publications and source records attributed to Weiyong He.
We prove that there is a unique $p_0\in [0,1)$, which can be characterized by the eigenvalue of Hilbert operator related to a convex body, that the even $L^p$ Minkowski problem has a unique solution for $p\geq p_0$, and the uniqueness fails for infinitely many convex bodies if $p p_0$.
A hypersymplectic structure on a 4-manifold is a triple of symplectic forms for which any non-zero linear combination is again symplectic. In 2006, Donaldson conjectured that on a compact 4-manifold any hypersymplectic structure can be deformed through cohomologous hypersymplectic structures to a hyperk\"ahler triple. We prove this under the assumption that the initial structure is invariant under an effective $S^1$-action. In particular we show that the underlying 4-manifold is diffeomorphic to $\mathbb{T}^4$.
The purpose of this article is to study the (residual) Monge-Amp\`{e}re mass of a plurisubharmonic function with an isolated unbounded locus. A general decomposition formula is obtained under the Sasakian structure of the unit sphere. In complex dimension two, we obtain an $L^{1}$-apriori estimate on the complex Monge-Amp\`{e}re operator. This induces an upper-bound estimate on the residual mass, provided with the uniform directional Lipschitz continuity. As an application, the zero mass conjecture is confirmed, if the function further separates the circular direction in its alternating part.
A hypersymplectic structure on a 4-manifold is a triple $\omega_1, \omega_2, \omega_3$ of 2-forms for which every non-trivial linear combination $a^1\omega_1 + a^2 \omega_2 + a^3 \omega_3$ is a symplectic form. Donaldson has conjectured that when the underlying manifold is compact, any such structure is isotopic in its cohomolgy class to a hyperk\"ahler triple. We prove this conjecture for a hypersymplectic structure on $T^4$ which is invariant under the standard $T^3$ action. The proof uses the hypersymplectic flow, a geometric flow which attempts to deform a given hypersymplectic structure to a hyperk\"ahler triple. We prove that on $T^4$, when starting from a $T^3$-invariant hypersymplectic structure, the flow exists for all time and converges modulo diffeomorphisms to the unique cohomologous hyperk\"ahler structure.
The aim of this article is to study the residual Monge-Ampère mass of a plurisubharmonic function with an isolated singularity, provided with the circular symmetry. With the aid of Sasakian geometry, we obtain an estimate on the residual mass of this function with respect to its Lelong number and maximal directional Lelong number. This result partially answers the zero mass conjecture raised by Guedj and Rashkovskii.
Given a symplectic class $[\omega]$ on a four torus $T^4$ (or a $K3$ surface), a folklore problem in symplectic geometry is whether symplectic forms in $[\omega]$ are isotropic to each other. We introduce a family of nonlinear Hodge heat flows on compact symplectic four manifolds to approach this problem, which is an adaption of nonlinear Hodge theory in symplectic geometry. As a particular example, we study a conformal Hodge heat flow in detail. We prove a stability result of the flow near an almost Kahler structure $(M, \omega, g)$. We also prove that, if $|\nabla \log u|$ stays bounded along the flow, then the flow exists for all time for any initial symplectic form $\rho\in [\omega]$ and it converges to $\omega$ smoothly along the flow with uniform control, where $u$ is the volume potential of $\rho$.
We introduce the notion of \emph{biharmonic almost complex structure} on a compact almost Hermitian manifold and we study its regularity and existence in dimension four. First we show that there always exist smooth energy-minimizing biharmonic almost complex structures for any almost Hermitian structure on a compact almost complex four manifold, and all energy-minimizers form a compact set. Then we study the existence problem when the homotopy class of an almost complex structure is specified. We obtain existence of energy-minimizing biharmonic almost complex structures which depends on the topology of $M^4$. When $M$ is simply-connected and non-spin, then for each homotopy class which is uniquely determined by its first Chern class, there exists an energy-minimizing biharmonic almost complex structure. When $M$ is simply-connected and spin, for each first Chern class, there are exactly two homotopy classes corresponding to the first Chern class. Given a homotopy class $[τ]$ of an almost complex structure, there exists a canonical operation on the homotopy classes $p$ satisfying $p^2=\text{id}$ such that $p([τ])$ and $[τ]$ have the same first Chern class. We prove that there exists an energy-minimizing biharmonic almost complex structure in (at least) one of the two homotopy classes, $[τ]$ and $p([τ])$. In general if $M$ is not necessarily simply-connected, we prove that there exists an energy-minimizing biharmonic almost complex structure in (at least) one of the two homotopy classes $[τ]$ and $p([τ])$. The study of biharmonic almost complex structures should have many applications, in particular for the smooth topology of the underlying almost complex four manifold. We briefly discuss an approach by considering the moduli space of biharmonic almost complex structures and propose a conjecture.
We prove that for any two closed Riemannian manifolds $M^{2m}$ ($m\geq 1$) and $N$, there exists a minimizing (extrinsic) $m$-polyharmonic map for every free homotopy class in $[M^{2m}, N]$, provided that the homotopy group $π_{2m}(N)$ is trivial. This generalizes the celebrated existence results for harmonic maps and biharmonic maps. We also prove that there exists a non-constant smooth polyharmonic map from $\mathbb{R}^{2m}$ to $N$ by a blowup analysis at an energy-concentration point for an energy-minimizing sequence if the convergence fails to be strong.
In this paper we consider the existence and regularity of weakly polyharmonic almost complex structures on a compact almost Hermitian manifold $M^{2m}$. Such objects satisfy the elliptic system weakly $[J, Δ^m J]=0$. We prove a very general regularity theorem for semilinear systems in critical dimensions (with \emph{critical growth nonlinearities}). In particular we prove that weakly biharmonic almost complex structures are smooth in dimension four.
The Gursky-Streets equation are introduced as the geodesic equation of a metric structure in conformal geometry. This geometric structure has played a substantial role in the proof of uniqueness of $σ_2$ Yamabe problem in dimension four. In this paper we solve the Gursky-Streets equations with uniform $C^{1, 1}$ estimates for $2k\leq n$. An important new ingredient is to show the concavity of the operator which holds for all $k\leq n$. Our proof of the concavity heavily relies on Garding's theory of hyperbolic polynomials and results from the theory of real roots for (interlacing) polynomials. Together with this concavity, we are able to solve the equation with the uniform $C^{1, 1}$ \emph{a priori estimates} for all the cases $n\geq 2k$. Moreover, we establish the uniqueness of the solution to the degenerate equations for the first time. As an application, we prove that if $k\geq 3$ and $M^{2k}$ is conformally flat, any solution solution of $σ_k$ Yamabe problem is conformal diffeomorphic to the round sphere $S^{2k}$.
We define and study the harmonic heat flow for almost complex structures which are compatible with a Riemannian structure $(M, g)$. This is a tensor-valued version of harmonic map heat flow. We prove that if the initial almost complex structure $J$ has small energy (depending on the norm $|\nabla J|$), then the flow exists for all time and converges to a Kähler structure. We also prove that there is a finite time singularity if the initial energy is sufficiently small but there is no Kähler structure in the homotopy class. A main technical tool is a version of monotonicity formula, similar as in the theory of the harmonic map heat flow. We also construct an almost complex structure on a flat four tori with small energy such that the harmonic heat flow blows up at finite time with such an initial data.
We study the existence and regularity of energy-minimizing harmonic almost complex structures. We have proved results similar to the theory of harmonic maps, notably the classical results of Schoen-Uhlenbeck and recent advance by Cheeger-Naber.
The scalar curvature equation for rotation invariant Kähler metrics on $\mathbb{C}^n \backslash \{0\}$ is reduced to a system of ODEs of order 2. By solving the ODEs, we obtain complete lists of rotation invariant zero or positive csck on $\mathbb{C}^n \backslash \{0\}$ in lower dimensions. We also prove that there does not exist negative csck on $\mathbb{C}^n \backslash \{0\}$ for $n=2,3$.
We extend profound results in pluripotential theory on Kahler manifolds to Sasaki setting via its transverse Kahler structure. As in Kahler case, these results form a very important piece to solve the existence of Sasaki metrics with constant scalar curvature (cscs) in terms of properness of K-energy. One main result is to generalize T. Darvas' theory on the geometric structure of the space of Kahler potentials in Sasaki setting. Along the way we extend most of corresponding results in pluripotential theory to Sasaki setting via its transverse Kahler structure.
In this paper we consider a class of fully nonlinear equations which cover the equation introduced by S. Donaldson a decade ago and the equation introduced by Gursky-Streets recently. We solve the equation with uniform weak $C^2$ estimates, which hold for degenerate case.
We study (transverse) scalar curvature type equation on compact Sasaki manifolds, in view of recent breakthrough of Chen-Cheng \cite{CC1, CC2, CC3} on existence of Kähler metrics with constant scalar curvature (csck) on compact Kähler manifolds. Following their strategy, we prove that given a Sasaki structure (with Reeb vector field and complex structure on its cone fixed ), there exists a Sasaki structure with transverse constant scalar curvature (cscs) if and only if the $\mathcal{K}$-energy is reduced proper modulo the identity component of the automorphism group which preserves both the Reeb vector field and transverse complex structure. Technically, the proof mainly consists of two parts. The first part is a priori estimates for scalar curvature type equations which are parallel to Chen-Cheng's results in \cite{CC2, CC3} in Sasaki setting. The second part is geometric pluripotential theory on a compact Sasaki manifold, building up on profound results in geometric pluripotential theory on Kähler manifolds. There are notable, and indeed subtle differences in Sasaki setting (compared with Kähler setting) for both parts (PDE and pluripotential theory). The PDE part is an adaption of deep work of Chen-Cheng \cite{CC1, CC2, CC3} to Sasaki setting with necessary modifications. While the geometric pluripotential theory on a compact Sasaki manifold has new difficulties, compared with geometric pluripotential theory in Kähler setting which is very intricate. We shall present the details of geometric pluripotential on Sasaki manifolds in a separate paper \cite{HL} (joint work with Jun Li).
In this paper, we study semilinear elliptic systems with critical nonlinearity of the form \begin{equation}\label{sys01} Δu=Q(x, u, \nabla u), \end{equation} for $u: \mathbb{R}^n\rightarrow \mathbb{R}^K$, $Q$ has quadratic growth in $\nabla u$. Our work is motivated by elliptic systems for harmonic map and biharmonic map. When $n=2$, such a system does not have smooth regularity in general for $W^{1, 2}$ weak solutions, by a well-known example of J. Frehse. Classical results of harmonic map, proved by F. Hélein (for $n=2$) and F. Béthuel (for $n\geq 3$), assert that a $W^{1, n}$ weak solution of harmonic map is always smooth. We extend Béthuel's result to above general system, that a $W^{1, n}$ weak solution of above system is smooth for $n\geq 3$. For a fourth order semilinear elliptic system with critical nonlinearity which extends biharmonic map, we prove a similar result, that a $W^{2, n/2}$ weak solution of such system is always smooth, for $n\geq 5$. We also construct various examples, and these examples show that our regularity results are optimal in various sense.
In this paper we extend recent breakthrough of Chen-Cheng \cite{CC1, CC2, CC3} on existence of constant scalar Kähler metric on a compact Kähler manifold to Calabi's extremal metric. Our argument follows \cite{CC3} and there are no new a prior estimates needed, but rather there are necessary modifications adapted to the extremal case. We prove that there exists an extremal metric with extremal vector $V$ if and only if the modified Mabuchi energy is proper, modulo the action the subgroup in the identity component of automorphism group which commutes with the flow of $V$. We introduce two essentially equivalent notions, called \emph{reductive properness} and \emph{reduced properness}. We observe that one can test reductive properness/reduced properness only for invariant metrics. We prove that existence of an extremal metric is equivalent to reductive properness/reduced properness of the modified Mabuchi energy.