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Zhizhang Wang

Publications and source records attributed to Zhizhang Wang.

At least 19 recordsLinked to original sources
arXiv2026

Inverse Hessian Curvature Flow in Minkowski Space II: The Dirichlet problem at infinity

This paper studies self-shrinkers and the long-time behavior of the inverse $σ_k$ curvature flow for noncompact entire spacelike strictly convex hypersurfaces in Minkowski space. In contrast to the co-compact setting, where the corresponding self-shrinker is rigid, the noncompact problem admits a rich family of self-shrinkers determined by their asymptotic data. More precisely, we formulate the self-shrinker equation as a fully nonlinear Dirichlet problem on hyperbolic space with prescribed data on its ideal boundary, and prove that every continuous negative boundary value determines a unique entire spacelike strictly convex self-shrinker. We further study the inverse $σ_k$ curvature flow starting from an entire spacelike strictly convex hypersurface satisfying a subsolution condition at infinity and a uniform positive lower bound for its $σ_k$ curvature. We prove global existence of the flow and show that the normalized flow converges locally smoothly to the unique self-shrinker with the same prescribed boundary value at infinity.

math.DG↗
arXiv2025

Second boundary value problem for the Hessian curvature flow

We investigate the evolution of strictly convex hypersurfaces driven by the $k$-Hessian curvature flow, subject to the second boundary condition. We first explore the translating solutions corresponding to this boundary value problem. Next, we establish the long-time existence of the flow and prove that it converges to a translating solution. To overcome the difficulty of driving boundary $C^2$ estimates, we employ an orthogonal invariance technique. Using this method, we extend the results of Schnürer-Smoczyk \cite{Schnurer2003} and Schnürer \cite{Schnurer2002} from the second boundary value problem of Gauss curvature flow to $k$-Hessian curvature flow.

math.AP↗
arXiv2025

Exciting games and Monge-Ampère equations

We consider a competition between $d+1$ players, and aim to identify the "most exciting game'' of this kind. This is translated, mathematically, into a stochastic optimization problem over martingales that live on the $d$-dimensional subprobability simplex $Δ$ and terminate on the vertices of $Δ$ (so-called win-martingales), with a cost function related to a scaling limit of Shannon entropies. We uncover a surprising connection between this problem and the seemingly unrelated field of Monge-Ampère equations: If $g$ solves \begin{equation*} \begin{cases} g(x)=\log \det\left(\frac{1}{2}\nabla^2 g(x)\right), \quad \, \ \ \ \ \, \, \, \, x \in Δ, \\ g(x)=\infty, \quad \quad \quad \quad \ \ \ \ \ \quad \quad \, \ \ \ \ \ x\in \partial Δ, \end{cases} \end{equation*} then the winning-probability of the players in the most exciting game is described by $$dM_s=\sqrt{\frac{2 (\nabla^2 g(M_s))^{-1}}{1-s} } \, dB_s.$$ To formalize this, a detailed quantitative analysis of the Monge-Ampère equation for $g$ is crucial. This is then leveraged to prove that $M$ is indeed an optimal win-martingale.

math.PR↗
arXiv2023

Gigahertz-rate-switchable wavefront shaping through integration of metasurfaces with photonic integrated circuit

Achieving spatiotemporal control of light at high-speeds presents immense possibilities for various applications in communication, computation, metrology, and sensing. The integration of subwavelength metasurfaces and optical waveguides offers a promising approach to manipulate light across multiple degrees of freedom at high-speed in compact photonic integrated circuit (PICs) devices. Here, we demonstrate a gigahertz-rate-switchable wavefront shaping by integrating metasurface, lithium niobite on insulator (LNOI) photonic waveguide and electrodes within a PIC device. As proofs of concept, we showcase the generation of a focus beam with reconfigurable arbitrary polarizations, switchable focusing with lateral focal positions and focal length, orbital angular momentum light beams (OAMs) as well as Bessel beams. Our measurements indicate modulation speeds of up to gigahertz rate. This integrated platform offers a versatile and efficient means of controlling light field at high-speed within a compact system, paving the way for potential applications in optical communication, computation, sensing, and imaging.

physics.optics↗
arXiv2023

Fast-speed and low-power-consumption optical phased array based on thin-film lithium niobate platform

Fast scanning-speed and low-power-consumption are becoming progressively more and more important in realizing high-performance chiplet optical phased arrays (OPAs). Here, we establish an integrated OPA based on thin-film lithium niobate-on-insulator (LNOI) platform to access these outstanding performances. Significantly, a lithium niobate (LN) OPA chip is implemented by 32/48 channels LN waveguides enabled by electro-optic modulations, which showcases the low power consumption (1.11nJ/π}) and fast operation speed (14.4 ns) promising the advantage of the LNOI platform for integrated OPAs. As results, we experimentally achieved a beam steering with a 62.2°*8.8° field of view (FOV) and a beam divergence of 2.4°*1.2°. Moreover, by employing sparse aperiodic arrays in waveguides design we obtained a significant reduction of lateral divergence to 0.33° for the radiation beam. This work demonstrate that remarkable advantage of LNOI platform for power-saving and scalable OPA chips for various applications.

physics.optics↗
arXiv2022

Entire $σ_k$ curvature flow in Minkowski space

In this paper, we study the $σ_k$ curvature flow of noncompact spacelike hypersurfaces in Minkowski space. We prove that if the initial hypersurface satisfies certain conditions, then the flow exists for all time. Moreover, we show that after rescaling, the flow converges to a self-expander.

math.DG↗
arXiv2022

Second order estimates for convex solutions of degenerate $k$-Hessian equations

The $C^{1,1}$ estimate of the Dirichlet problem for degenerate $k$-Hessian equations with non-homogenous boundary conditions is an open problem, if the right hand side function $f$ is only assumed to satisfy $f^{1/(k-1)} \in C^{1,1}$. In this paper, we solve this problem for convex solutions defined in the strictly convex bounded domain.

math.AP↗
arXiv2022

Entire convex curvature flow in Minkowski space

In this paper, we study fully nonlinear curvature flows of noncompact spacelike hypersurfaces in Minkowski space. We prove that if the initial hypersurface satisfies certain conditions, then the flow exists for all time. Moreover, we show that after rescaling the flow converges to the future timelike hyperboloid, which is a self-expander.

math.DG↗
arXiv2022

Entire self-expanders for power of $σ_k$ curvature flow in Minkowski space

In [19], we prove that if an entire, spacelike, convex hypersurface $\mathcal{M}_{u_0}$ has bounded principal curvatures, then the $σ_k^{1/α}$ (power of $σ_k$) curvature flow starting from $\mathcal{M}_{u_0}$ admits a smooth convex solution $u$ for $t>0.$ Moreover, after rescaling, the flow converges to a convex self-expander $\tilde{\mathcal{M}}=\{(x, \tilde{u}(x))\mid x\in\mathbb{R}^n\}$ that satisfies $σ_k(κ[\tilde{\mathcal{M}}])=(-\left )^α.$ Unfortunately, the existence of self-expander for power of $σ_k$ curvature flow in Minkowski space has not been studied before. In this paper, we fill the gap.

math.DG↗
arXiv2021

Entire spacelike constant $σ_k$ curvature hypersurfaces with prescribed boundary data at infinity

In this paper, we investigate the existence and uniqueness of convex, entire, spacelike hypersurfaces of constant $σ_k$ curvature with prescribed set of lightlike directions $\mathcal{F}\subset\mathbb{S}^{n-1}$ and perturbation $q$ on $\mathcal{F}$. We prove that given a closed set $\mathcal{F}$ in the ideal boundary at infinity of hyperbolic space and a perturbation $q$ that satisfies some mild conditions, there exists a complete entire spacelike constant $σ_k$ curvature hypersurface $\mathcal{M}_u$ with prescribed set of lightlike directions $\mathcal{F}$ satisfying when $\frac{x}{|x|}\in\mathcal{F},$ as $|x|\rightarrow\infty,$ $u(x)-|x|\rightarrow q\left(\frac{x}{|x|}\right).$ This result is new even for the case of constant Gauss curvature. We also prove that when the Gauss map image is a half disc $\bar{B}_1^+$ and the perturbation $q\equiv 0,$ if a CMC hypersurface $\mathcal{M}_u$ satisfies $|u(x)-V_{\bar{\mathcal{B}}_+}(x)|$ is bounded, then $u(x)$ is unique.

math.DG↗
arXiv2020

The prescribed curvature problem for entire hypersurfaces in Minkowski space

We prove three results in this paper. First, we prove for a wide class of functions $φ\in C^2(\mathbb{S}^{n-1})$ and $ψ(X, ν)\in C^2(\mathbb{R}^{n+1}\times\mathbb{H}^n),$ there exists a unique, entire, strictly convex, spacelike hypersurface $M_u$ satisfying $σ_k(κ[M_u])=ψ(X, ν)$ and $u(x)\rightarrow |x|+φ\left(\frac{x}{|x|}\right)$ as $|x|\rightarrow\infty.$ Second, when $k=n-1, n-2,$ we show the existence and uniqueness of entire, $k$-convex, spacelike hypersurface $M_u$ satisfying $σ_k(κ[M_u])=ψ(x, u(x))$ and $u(x)\rightarrow |x|+φ\left(\frac{x}{|x|}\right)$ as $|x|\rightarrow\infty.$ Last, we obtain the existence and uniqueness of entire, strictly convex, downward translating solitons $M_u$ with prescribed asymptotic behavior at infinity for $σ_k$ curvature flow equations. Moreover, we prove that the downward translating solitons $M_u$ have bounded principal curvatures.

math.DG↗
arXiv2020

Entire spacelike hypersurfaces with constant $σ_k$ curvature in Minkowski space

In this paper, we prove the existence of smooth, entire, strictly convex, spacelike, constant $σ_k$ curvature hypersurfaces with prescribed lightlike directions in Minkowski space. This is equivalent to prove the existence of smooth, entire, strictly convex, spacelike, constant $σ_k$ curvature hypersurfaces with prescribed Gauss map image. We also show that there doesn't exist any entire, convex, strictly spacelike, constant $σ_k$ curvature hypersurfaces. Moreover, we generalize the result in \cite{RWX} and construct strictly convex, spacelike, constant $σ_k$ curvature hypersurface with bounded principal curvature, whose image of the Gauss map is the unit ball.

math.DG↗
arXiv2020

Entire spacelike hypersurfaces with constant $σ_{n-1}$ curvature in Minkowski space

We prove that, in Minkowski space, if a spacelike, $(n-1)$-convex hypersurface $M$ with constant $σ_{n-1}$ curvature has bounded principal curvatures, then $M$ is convex. Moreover, if $M$ is not strictly convex, after an $\mathbb{R}^{n,1}$ rigid motion, $M$ splits as a product $M^{n-1}\times\mathbb{R}.$ We also construct nontrivial examples of strictly convex, spacelike hypersurface $M$ with constant $σ_{n-1}$ curvature and bounded principal curvatures.

math.DG↗
arXiv2020

Notes on the curvature estimates for Hessian equations

The main result of this paper gives a plenary proof on the curvature estimates for $k$ curvature equations with general right hand sides with $n<2k$ based on a concavity inequality. We further give a explicit lower bound of the inequality.

math.AP↗