The Fu-Yau equation in higher dimensions
In this paper, we prove the existence of solutions to the Fu-Yau equation on compact Kähler manifolds. As an application, we give a class of non-trivial solutions of the modified Strominger system.
math.DG↗
arXiv subjects
Jianchun Chu
Publications and source records attributed to Jianchun Chu.
Optimal regularity of plurisubharmonic envelopes on compact Hermitian manifolds
In this paper, we prove the $C^{1, 1}$-regularity of the plurisubharmonic envelope of a $C^{1,1}$ function on a compact Hermitian manifold. We also present examples to show this regularity is sharp.
math.AP↗
$C^{1,1}$ regularity for degenerate complex Monge-Ampère equations and geodesic rays
We prove a $C^{1,1}$ estimate for solutions of complex Monge-Ampère equations on compact Kähler manifolds with possibly nonempty boundary, in a degenerate cohomology class. This strengthens previous estimates of Phong-Sturm. As applications we deduce the local $C^{1,1}$ regularity of geodesic rays in the space of Kähler metrics associated to a test configuration, as well as the local $C^{1,1}$ regularity of quasi-psh envelopes in nef and big classes away from the non-Kähler locus.
math.DG↗
The Monge-Ampère equation for non-integrable almost complex structures
We show existence and uniqueness of solutions to the Monge-Ampere equation on compact almost complex manifolds with non-integrable almost complex structure.
math.AP↗
On the $C^{1,1}$ regularity of geodesics in the space of Kähler metrics
We prove that any two Kahler potentials on a compact Kahler manifold can be connected by a geodesic segment of C^{1,1} regularity. This follows from an a priori interior real Hessian bound for solutions of the nondegenerate complex Monge-Ampere equation, which is independent of a positive lower bound for the right hand side.
math.DG↗
The $2$-nd Hessian type equation on almost Hermitian manifolds
In this paper, we derive the second order estimate to the $2$-nd Hessian type equation on a compact almost Hermitian manifold.
math.AP↗
Quantitative stratification of $F$-subharmonic functions
In this paper, we study the singular sets of $F$-subharmonic functions $u: B_{2}(0^{n})\rightarrow\mathbf{R}$, where $F$ is a subequation. The singular set $\mathcal{S}(u)\subset B_{2}(0^{n})$ has a stratification $\mathcal{S}^{0}(u)\subset\mathcal{S}^{1}(u)\subset\cdots\subset\mathcal{S}^{k}(u)\subset\cdots\subset\mathcal{S}(u)$, where $x\in\mathcal{S}^{k}(u)$ if no tangent function to $u$ at $x$ is $(k+1)$-homogeneous. We define the quantitative stratification $\mathcal{S}_{η,r}^{k}(u)$ and $\mathcal{S}_η^{k}(u)=\cap_{r}\mathcal{S}_{η,r}^{k}(u)$. When homogeneity of tangents holds for $F$, we prove that $dim_{H}\mathcal{S}^{k}(u)\leq k$ and $\mathcal{S}(u)=\mathcal{S}^{n-p}(u)$, where $p$ is the Riesz characteristic of $F$. And for the top quantitative stratification $\mathcal{S}_η^{n-p}(u)$, we have the Minkowski estimate $\text{Vol}(B_{r}(\mathcal{S}_η^{n-p}(u)\cap B_{1}(0^{n})))\leq Cη^{-1}(\int_{B_{1+r}(0^{n})}Δu)r^{p}$. When uniqueness of tangents holds for $F$, we show that $S_η^{k}(u)$ is $k$-rectifiable, which implies $\mathcal{S}^{k}(u)$ is $k$-rectifiable. When strong uniqueness of tangents holds for $F$, we introduce the monotonicity condition and the notion of $F$-energy. By using refined covering argument, we obtain a definite upper bound on the number of $\{Θ(u,x)\geq c\}$ for $c>0$, where $Θ(u,x)$ is the density of $F$-subharmonic function $u$ at $x$. Geometrically determined subequations $F(\mathbb{G})$ is a very important kind of subequation (when $p=2$, homogeneity of tangents holds for $F(\mathbb{G})$; when $p>2$, uniqueness of tangents holds for $F(\mathbb{G})$). By introducing the notion of $\mathbb{G}$-energy and using quantitative differentation argument, we obtain the Minkowski estimate of quantitative stratification $\text{Vol}(B_{r}(\mathcal{S}_{η,r}^{k}(u))\cap B_{1}(0^{n}))\leq Cr^{n-k-η}$.
math.AP↗
The parabolic Monge-Ampère equation on compact almost Hermitian manifolds
We prove the long time existence and uniqueness of solutions to the parabolic Monge-Ampère equation on compact almost Hermitian manifolds. We also show that the normalization of solution converges to a smooth function in $C^{\infty}$ topology as $t\rightarrow\infty$. Up to scaling, the limit function is a solution of the Monge-Ampère equation. This gives a parabolic proof of existence of solutions to the Monge-Ampère equation on almost Hermitian manifolds.
math.AP↗
$C^{2,α}$ regularities and estimates for nonlinear elliptic and parabolic equations in geometry
We give sharp $C^{2,α}$ estimates for solutions of some fully nonlinear elliptic and parabolic equations in complex geometry and almost complex geometry, assuming a bound on the Laplacian of the solution. We also prove the analogous results to complex Monge-Ampère equations with conical singularities. As an application, we obtain a local estimate for Calabi-Yau equation in almost complex geometry. We also improve the $C^{2,α}$ regularities and estimates for viscosity solutions to some uniformly elliptic and parabolic equations. All our results are optimal regarding the Hölder exponent.
math.DG↗
The complex Monge-Ampère equation on some compact Hermitian manifolds
We consider the complex Monge-Ampère equation on compact manifolds when the background metric is a Hermitian metric (in complex dimension two) or a kind of Hermitian metric (in higher dimensions). We prove that the Laplacian estimate holds when $F$ is in $W^{1,q_{0}}$ for any $q_{0}>2n$. As an application, we show that, up to scaling, there exists a unique classical solution in $W^{3,q_{0}}$ for the complex Monge-Ampère equation when $F$ is in $W^{1,q_{0}}$.
math.DG↗