We establish first-and second-order gradient estimates for positive solutions of the heat equations under general geometric flows. Our results generalize the recent work of S. Liu, who established similar results for the Ricci flow. Both results can also be considered as the generalization of P. Li, S. T. Yau, and J. Li's gradient estimates under geometric flow setting. We also give an application to the mean curvature flow.

The skew mean curvature flow(SMCF), which origins from the study of fluid dynamics, describes the evolution of a codimension two submanifold along its binormal direction. We study the basic properties of the SMCF and prove the existence of a short-time solution to the initial value problem of the SMCF of compact surfaces in Euclidean space R 4 . A Sobolev-type embedding theorem for the second fundamental forms of two dimensional surfaces is also proved, which might be of independent interest.General SMCF naturally arises in higher dimensional hydrodynamics. A singular vortex in a fluid is called a vortex membrane if it is supported on a codimension two subset. The law of locally induced motion of a vortex membrane can by deduced from the Euler equation by applying the Biot-Savart formula. In 2012, Shashikanth [33] first investigated the motion of a vortex membrane in R 4 and showed that it is governed by the two dimensional SMCF. Khesin [21] generalized this conclusion to any dimensional vortex membranes in Euclidean spaces and gave the formal definition of the SMCF which we apply here.The SMCF also emerges in the study of asymptotic dynamics of vortices in the context of superfluidity and superconductivity. Here the model is usually given by a PDE of a complex field and the vortices are just zero sets of the solutions. For example, for the Ginzburg-Landau heat flow, it was shown that asymptotically the energy concentrates on the codimension two vortices which moves along the mean curvature flow [26,19,3,20]. Similar phenomena are observed for wave and Schrödinger type PDEs. In particular, for the Gross-Pitaevskii equation which models the wave function associated with a Bose-Einstein condensate, physics evidences indicate that the vortices would evolve along the SMCF. This was first verified by Lin [29] for the vortex filaments in three space dimensions. For higher dimensions, Jerrard [17] proved this conjecture when the initial singular set is a codimension two sphere with multiplicity one in 2002. It is worth mentioning that he also proposed a notion of weak solution to the SMCF.Besides its physical significance, the SMCF is a rather canonical geometric flow for codimension two submanifolds which can be viewed as the Schrödinger-type counterpart of the well-known mean curvature flow(MCF). In fact, the SMCF has a notable Hamiltonian structure and is volume preserving. The infinite dimensional space of codimension two immersions of a Riemannian manifold admits a generalized Marsden-Weinstein symplectic structure [30], and the SMCF turns out to be the Hamiltonian flow of the volume functional on this space. This fact was first noted by Haller and Vizman [12] where they studied the non-linear Grassmannians. For completeness, we include a detailed explanation in Section 2.1 below.The SMCF is also related to another important Hamiltonian flow, namely, the Schrödinger flow [8,39,7]. The Schrödinger flow stems from the study of ferromagnetism and is the Hamiltonian flow of the energy functional defined on the space ...

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The null curves on 3-null cone have the applications in the studying of horizon types. Via the pseudo-scalar product and Frenet equations, the differential geometry of null curves on 3-null cone is obtained. In the local sense, the curvature describes the contact of submanifolds with pseudo-spheres. We introduce the geometric properties of the null curves on 3-null cone and unit semi-Euclidean 3-spheres, respectively. On the other hand, we give the existence conditions of null Bertrand curves on 3-null cone and unit semi-Euclidean 3-spheres. c ⃝2015 All rights reserved.

In this article, we prove some nonexistence results for the translating solitons to the symplectic mean curvature flows or to the almost calibrated Lagrangian mean curvature flows under some curvature assumptions.

This paper mainly deals with the type II singularities of the mean curvature flow from a symplectic surface or from an almost calibrated Lagrangian surface in a Kähler surface. The relation between the maximum of the Kähler angle and the maximum of |H| 2 on the limit flow is studied. The authors also show the nonexistence of type II blow-up flow of a symplectic mean curvature flow which is normal flat or of an almost calibrated Lagrangian mean curvature flow which is flat.

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In this paper, we prove that if the area functional of a surface Σ 2 in a symplectic manifold (M 2n ,ω) has a critical point or has a compatible stable point in the same cohomology class, then it must be J-holomorphic. Inspired by a classical result of Lawson-Simons, we show how various restrictions of the stability assumption to variations of metrics in the space "projectively induced" metrics are enough to give the desired conclusion.Mathematics Subject Classification (2010): 53A10 (primary), 53D05 (secondary).In light of our knowledge about the relationship between stable minimal surfaces and holomorphic curves, it is natural to look at special properties of the second variation of the functional A:Note that the definition of A-stability (as well as all the other stability notions we are going to study) does not require ρ to be a critical point of the area functional. Our next result shows that the existence of a compatible stable point is also enough to guarantee the J-holomorphicity:Theorem 3.2: Let (M 2n ,ω, J) be a compact symplectic manifold with compatible almost complex structure J and F : Σ 2 → M be an immersion. If the area functional A has a compatible stable point ρ ∈ H, then the immersion is J-holomorphic.As above, the converse is also true even without assuming that J is compatible withω.Checking the proof of the above theorem carefully, we see that the result is also true in the complete noncompact case. In particular, this applies to minimal submanifold in R 2n . In this case, we need the test function to have compact support.Theorem 3.4: Let (M 2n ,ω, J) be a complete noncompact symplectic manifold with compatible almost complex structure J and F : Σ 2 → M be an immersion. If the area functional A has a compatible stable point ρ ∈ H, then the immersion is J-holomorphic.