Related papers: Remarks on the self-shrinking Clifford torus

We prove that all entire smooth strictly convex self-shrinking solutions on $\mathbb{R}^n$ to the Hessian quotient flows must be quadratic. This generalizes the rigidity theorem for entire self-shrinking solutions to the Lagrangian mean…

We show that the product of two round shrinking spheres is an isolated self-shrinker in any codimension, modulo rotations. Moreover we prove explicit {\L}ojasiewicz inequalities near such products. {\L}ojasiewicz inequalities were…

We prove regularity, global existence, and convergence of Lagrangian mean curvature flows in the two-convex case. Such results were previously only known in the convex case, of which the current work represents a significant improvement.…

Let f :S\to B be a non locally trivial fibred surface. We prove a lower bound for the slope of f depending increasingly from the relative irregularity of f and the Clifford index of the general fibres.

Combining the usual energy functional with a higher-order conserved quantity originating from integrability theory, we show that the black soliton is a local minimizer of a quantity that is conserved along the flow of the cubic defocusing…

In this paper, we investigate the Hamiltonian-stability of Lagrangian tori in the complex hyperbolic space $\mathbb{C}H^n$. We consider a standard Hamiltonian $T^n$-action on $\mathbb{C}H^n$, and show that every Lagrangian $T^n$-orbits in…

Rotating shear flows, when angular momentum increases and angular velocity decreases as functions of radiation coordinate, are hydrodynamically stable under linear perturbation. The Keplerian flow is an example of such systems which appears…

By the integral method we prove that any space-like entire graphic self-shrinking solution to Lagrangian mean curvature flow in $\R^{2n}_{n}$ with the indefinite metric $\sum_i dx_idy_i$ is flat. This result improves the previous ones in…

A concise review is given of astrophysically motivated experimental and theoretical research on Taylor-Couette flow. The flows of interest rotate differentially with inner cylinder faster than outer one but are linearly stable against…

Hamiltonian flows on compact surfaces are characterized, and the topological invariants of such flows with finitely many singular points are constructed from the viewpoints of integrable systems, fluid mechanics, and dynamical systems.…

We consider volume-preserving perturbations of the time-one map of the geodesic flow of a compact surface with negative curvature. We show that if the Liouville measure has Lebesgue disintegration along the center foliation then the…

In this article, the author investigates flow lines of the classical Willmore flow, which start to move in a smooth parametrization of a Hopf-torus in $\mathbb{S}^3$. We prove that any such flow line of the Willmore flow exists globally, in…

In this paper we propose a class of local definitions of weak lower scalar curvature bounds that is well defined for $C^0$ metrics. We show the following: that our definitions are stable under greater-than-second-order perturbation of the…

We study singularities of Lagrangian mean curvature flow in $\C^n$ when the initial condition is a zero-Maslov class Lagrangian. We start by showing that, in this setting, singularities are unavoidable. More precisely, we construct…

In this paper we study Lagrangian tori in ${\mathbb C}P^2$. A two-dimensional periodic Schr\"odinger operator is associated with every Lagrangian torus in ${\mathbb C}P^2$. We introduce an energy functional for tori as an integral of the…

We prove the ill-posedness in $ H^s(\T) $, $s<0$, of the periodic cubic Schr\"odinger equation in the sense that the flow-map is not continuous from $H^s(\T) $ into itself for any fixed $ t\neq 0 $. This result is slightly stronger than the…

Ejiri's torus in $S^5$ is the first example of Willmore surface which is not conformally equivalent to any minimal surface in any space forms. Li and Vrancken classified all Willmore surfaces of tensor product in $S^{n}$ by reducing them…

Huisken studied asymptotic behavior of a mean curvature flow in a Euclidean space when it develops a singularity of type I, and proved that its rescaled flow converges to a self-shrinker in the Euclidean space. In this paper, we generalize…

A well known result of Da Rios and Levi-Civita says that a closed planar curve is elastic if and only if it is stationary under the localized induction (or smoke ring) equation, where stationary means that the evolution under the localized…

The Clifford torus is a torus in a three-dimensional sphere. Homogeneous tori are simple generalization of the Clifford torus which still in a three-dimensional sphere. There is a way to construct tori in a three-dimensional sphere using…