Singularities of symplectic and Lagrangian mean curvature flows

In this paper we study the singularities of the mean curvature flow from a symplectic surface or from a Lagrangian surface in a K\"ahler-Einstein surface. We prove that the blow-up flow $\Sigma_s^\infty$ at a singular point $(X_0, T_0)$ of a symplectic mean curvature flow $\Sigma_t$ or of a Lagrangian mean curvature flow $\Sigma_t$ is a non trivial minimal surface in ${\bf R}^4$, if $\Sigma_{-\infty}^\infty$ is connected

The mean curvature flow along the K\"ahler-Ricci flow

Let $(M,\overline{g})$ be a K\"ahler surface, and $\Sigma$ an immersed surface in $M$. The K\"ahler angle of $\Sigma$ in $M$ is introduced by Chern-Wolfson \cite{CW}. Let $(M,\overline{g}(t))$ evolve along the K\"ahler-Ricci flow, and $\Sigma_t$ in $(M,\overline{g}(t))$ evolve along the mean curvature flow. We show that the K\"ahler angle $\alpha(t)$ satisfies the evolution equation: $$(\frac{\partial}{\partial t}-\Delta)\cos\alpha=|\overline\nabla J_{\Sigma_t}|^2\cos\alpha+R\sin^2\alpha\cos\alpha,$$ where $R$ is the scalar curvature of $(M, \overline{g}(t))$. The equation implies that, if the initial surface is symplectic (Lagrangian), then along the flow, $\Sigma_t$ is always symplectic (Lagrangian) at each time $t$, which we call a symplectic (Lagrangian) K\"ahler-Ricci mean curvature flow. In this paper, we mainly study the symplectic K\"ahler-Ricci mean curvature flow.Comment: 23 page

The second type singularity of symplectic and Lagrangian mean curvature flows

In this paper we mainly study the type II singularities of the mean curvature flow from a symplectic surface or from an almost calibrated Lagrangian surface in a K \"ahler-Einstein surface. We show the relation between the maximum of the K\"ahler angle and the maximum of $|H|^2$ on the limit flow

Global strong solution to the density-dependent incompressible flow of liquid crystals

The initial-boundary value problem for the density-dependent incompressible flow of liquid crystals is studied in a three-dimensional bounded smooth domain. For the initial density away from vacuum, the existence and uniqueness is established for both the local strong solution with large initial data and the global strong solution with small data. It is also proved that when the strong solution exists, a weak solution with the same data must be equal to the unique strong solution.Comment: arXiv admin note: substantial text overlap with arXiv:1108.547

The mean curvature flow approach to the symplectic isotopy problem

Let $M$ be a K\"ahler-Einstein surface with positive scalar curvature. If the initial surface is sufficiently close to a holomorphic curve, we show that the mean curvature flow has a global solution and it converges to a holomorphic curve.Comment: 7 page

Long time existence of the symplectic mean curvature flow

Let $(M,\bar{g})$ be a K\"ahler surface with a constant holomorphic sectional curvature $k>0$, and $\Sigma$ an immersed symplectic surface in $M$. Suppose $\Sigma$ evolves along the mean curvature flow in $M$. In this paper, we show that the symplectic mean curvature flow exists for long time and converges to a holomorphic curve if the initial surface satisfies $|A|^2\leq 2/3|H|^2+1/2 k$ and $\cos\alpha\geq \frac{\sqrt{30}}{6}$ or $|A|^2\leq 2/3 |H|^2+4/5 k\cos\alpha$ and $\cos\alpha\ge251/265$.Comment: 17page

On a SAV-MAC scheme for the Cahn-Hilliard-Navier-Stokes Phase Field Model

We construct a numerical scheme based on the scalar auxiliary variable (SAV) approach in time and the MAC discretization in space for the Cahn-Hilliard-Navier-Stokes phase field model, and carry out stability and error analysis. The scheme is linear, second-order, unconditionally energy stable and can be implemented very efficiently. We establish second-order error estimates both in time and space for phase field variable, chemical potential, velocity and pressure in different discrete norms. We also provide numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of the our scheme

Robust Robot-assisted Tele-grasping Through Intent-Uncertainty-Aware Planning

In teleoperation, research has mainly focused on target approaching, where we deal with the more challenging object manipulation task by advancing the shared control technique. Appropriately manipulating an object is challenging due to the fine motion constraint requirements for a specific manipulation task. Although these motion constraints are critical for task success, they often are subtle when observing ambiguous human motion. The disembodiment problem and physical discrepancy between the human and robot hands bring additional uncertainty, further exaggerating the complications of the object manipulation task. Moreover, there is a lack of planning and modeling techniques that can effectively combine the human and robot agents' motion input while considering the ambiguity of the human intent. To overcome this challenge, we built a multi-task robot grasping model and developed an intent-uncertainty-aware grasp planner to generate robust grasp poses given the ambiguous human intent inference inputs. With these validated modeling and planning techniques, it is expected to extend teleoperated robots' functionality and adoption in practical telemanipulation scenarios.Comment: 12 pages, 6 figures, journal pape

The deformation of symplectic critical surfaces in a K\"ahler surface-II---Compactness

In this paper we consider the compactness of $\beta$-symplectic critical surfaces in a K\"ahler surface. Let $M$ be a compact K\"ahler surface and $\Sigma_i\subset M$ be a sequence of closed $\beta_i$-symplectic critical surfaces with $\beta_i\to\beta_0\in (0,\infty)$. Suppose the quantity $\int_{\Sigma_i}\frac{1}{\cos^q\alpha_i}d\mu_i$ (for some $q>4$) and the genus of $\Sigma_{i}$ are bounded, then there exists a finite set of points ${\mathcal S}\subset M$ and a subsequence $\Sigma_{i'}$ that converges uniformly in the $C^l$ topology (for any $l<\infty$) on compact subsets of $M\backslash {\mathcal S}$ to a $\beta_0$-symplectic critical surface $\Sigma\subset M$, each connected component of $\Sigma\setminus {\mathcal S}$ can be extended smoothly across ${\mathcal S}$.Comment: 23 page

Estimates for Parametric Marcinkiewicz Integrals on Musielak-Orlicz Hardy Spaces

Let $\varphi:\mathbb{R}^n\times[0,\,\infty) \rightarrow [0,\,\infty)$ satisfy that $\varphi(x,\,\cdot)$, for any given $x\in\mathbb{R}^n$, is an Orlicz function and $\varphi(\cdot\,,t)$ is a Muckenhoupt $A_\infty$ weight uniformly in $t\in(0,\,\infty)$. The Musielak-Orlicz Hardy space $H^\varphi(\mathbb{R}^n)$ generalizes both of the weighted Hardy space and the Orlicz Hardy space and hence has a wide generality. In this paper, the authors first prove the completeness of both of the Musielak-Orlicz space $L^\varphi(\mathbb{R}^n)$ and the weak Musielak-Orlicz space $WL^\varphi(\mathbb{R}^n)$. Then the authors obtain two boundedness criterions of operators on Musielak-Orlicz spaces. As applications, the authors establish the boundedness of parametric Marcinkiewicz integral $\mu^\rho_\Omega$ from $H^\varphi(\mathbb{R}^n)$ to $L^\varphi(\mathbb{R}^n)$ (resp. $WL^\varphi(\mathbb{R}^n)$) under weaker smoothness condition (resp. some Lipschitz condition) assumed on $\Omega$. These results are also new even when $\varphi(x,\,t):=\phi(t)$ for all $(x,\,t)\in\mathbb{R}^n\times[0,\,\infty)$, where $\phi$ is an Orlicz function.Comment: 30 pages, accepted by JM