Related papers: The harmonic heat flow of almost complex structure…

Let $B_1$ be the unit open disk in $\Real^2$ and $M$ be a closed Riemannian manifold. In this note, we first prove the uniqueness for weak solutions of the harmonic map heat flow in $H^1([0,T]\times B_1,M)$ whose energy is non-increasing in…

The conformal heat flow of harmonic maps is a system of evolution equations combined with harmonic map flow with metric evolution in conformal direction. It is known that global weak solution of the flow exists and smooth except at mostly…

In this work, we obtain a short time existence result for harmonic map heat flow coupled with a smooth family of complete metrics in the domain manifold. Our results generalize short time existence results for harmonic map heat flow by…

Given a symplectic class $[\omega]$ on a four torus $T^4$ (or a $K3$ surface), a folklore problem in symplectic geometry is whether symplectic forms in $[\omega]$ are isotropic to each other. We introduce a family of nonlinear Hodge heat…

In this paper, we first establish regularity of the heat flow of biharmonic maps into the unit sphere $S^L\subset\mathbb R^{L+1}$ under a smallness condition of renormalized total energy. For the class of such solutions to the heat flow of…

We adopt the Koch-Tataru theory for the Navier-Stokes equations, based on Carleson measure estimates, to develop a scaling-critical low-regularity framework for half-harmonic map heat flows. This nonlocal variant of the harmonic map heat…

In this paper, we establish the uniqueness of heat flow of harmonic maps into (N, h) that have sufficiently small renormalized energies, provided that N is either a unit sphere $S^{k-1}$ or a compact Riemannian homogeneous manifold without…

In this article, we study the the harmonic map heat flow from a manifold with conic singularities to a closed manifold. In particular, we have proved the short time existence and uniqueness of solutions as well as the existence of global…

We formulate the gradient Dirichlet flow of $Sp(2)Sp(1)$-structures on $8$-manifolds, as the first systematic study of a geometric quaternion-K\"ahler (QK) flow. Its critical condition of \emph{harmonicity} is especially relevant in the QK…

We introduce and study a conformal heat flow of harmonic maps defined by an evolution equation for a pair consisting of a map and a conformal factor of metric on the two-dimensional domain. This flow is designed to postpone finite time…

Let $\Sigma$ be a compact oriented surface and $N$ a compact K\"ahler manifold with nonnegative holomorphic bisectional curvature. For a solution of harmonic map flow starting from an almost-holomorphic map $\Sigma \to N$ (in the energy…

We show an energy convexity along any harmonic map heat flow with small initial energy and fixed boundary data on the unit 2-disk. In particular, this gives an affirmative answer to a question raised by W. Minicozzi asking whether such…

In this paper, we introduce the regularized conformal heat flow of $n$-harmonic maps, or simply regularized $n$-conformal heat flow from $n$-dimensional Riemannian manifold. This is a system of evolution equations combined with regularized…

We consider the harmonic heat flow for maps from a compact Riemannian manifold into a Riemannian manifold that is complete and of non-positive curvature. We prove that if the harmonic heat flow converges to a limiting harmonic map that is a…

In this paper, we consider the pseudoharmonic heat flow with small initial horizontal energy and give the existence of pseudoharmonic maps from closed pseudo-Hermitian manifolds to closed Riemannian manifolds.

We study the existence and regularity of energy-minimizing harmonic almost complex structures. We have proved results similar to the theory of harmonic maps, notably the classical results of Schoen-Uhlenbeck and recent advance by…

Let $(M,J,\Omega)$ be a closed polarized complex manifold of K\"ahler type. Let $G$ be the maximal compact subgroup of the automorphism group of $(M,J)$. On the space of K\"ahler metrics that are invariant under $G$ and represent the…

For any $k$-dimensional smooth, compact Riemannian manifold $(N, h)\subset\mathbb R^L$ without boundary, there exists an $\varepsilon_0>0$ such that for any homogeneous of degree zero map $u_0(x)=\phi_0(\frac{x}{|x|}):\mathbb R^n\to N$…

Given a $7$-dimensional compact Riemannian manifold $\left( M,g\right) $ that admits $G_{2}$-structure, all the $G_{2}$-structures that are compatible with the metric $g$ are parametrized by unit sections of an octonion bundle over $M$. We…

We investigate the harmonic map heat flow from a compact Riemannian manifold \( M \) into the moduli space \( \mathcal{M}_1 \) of unit-area flat tori, which carries a natural hyperbolic structure as the quotient \( \mathrm{SL}(2,\mathbb{Z})…