arXiv subjects
Publications and source records attributed to Woongbae Park.
In this paper we introduce conformal heat flow of (extrinsic) biharmonic maps on $4$-manifold, simply called bi-conformal heat flow (bi-CHF), and study its properties. Similar to other CHF of harmonic maps and regularized $n$-harmonic maps, (CHF and regularized $n$-CHF respectively), we obtain global smoothness and no finite time singularity.
For elliptic systems defined on Riemann surfaces, Liouville and Toda systems represent two well-known classes exhibiting drastically different solution structures. Over the years, existence results for these systems have highlighted discrepancies due to their unique solution structures. In this work, we aim to construct a monotone entropy form and establish the long-term existence of a flow of parabolic systems. As a result of our main theorem, we can prove existence results for some broad classes of elliptic systems, including both Liouville and Toda systems. The strength of our results is further underscored by the fact that no topological information about the Riemann surfaces is required and no positive lower bound of coefficient functions is postulated.
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 finitely many singular points. In this paper, we show that no finite time singularity occurs, unlike the usual harmonic map flow. And if the initial energy is small, we can obtain the uniform convergence of the map to a point and the conformal factor of the metric under some time sequence $t_n \to \infty$. Also, under the assumption that energy concentration is uniform in time, we show that there exists a sequence of time $t_n \to \infty$ such that $f(\cdot,t_n)$ converges to a harmonic map in $W^{1,2}$ on any compact set away from at most finitely many points.
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 $n$-harmonic map flow and a metric evolution equation in conformal direction. For $n=2$, the conformal heat flow does not develop finite time singularity unlike usual harmonic map flow \cite{P23} (Park, 2024). In this paper, we show the analogous result, that regularized $n$-conformal heat flow does not develop finite time singularity unlike the (regularized) $n$-harmonic map flow.
We show that for any $k$ and $s > \frac{k+1}{k+2}$ there exist neither $W^{s,\frac{k}{s}}$-Sobolev nor $C^s$-Hölder homeomorphisms from the disk $\mathbb{B}^n$ into $\mathbb{R}^N$ whose gradient has rank $< k$ in distributional sense. This complements known examples of such kind of homeomorphisms whose gradient has rank $<k$ almost everywhere.
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 singularity but does not get rid of possibility of bubble forming. We show that Struwe type global weak solution exists, which is smooth except at most finitely many points.
Let $\mathcal{N} \subset \mathbb{R}^M$ be a smooth simply connected compact manifold without boundary. A rational homotopy subgroup of $π_{N}(\mathcal{N})$ is represented by a homomorphism \[{\rm deg}: π_{N}(\mathcal{N}) \to \mathbb{R}.\] For maps $f: \mathbb{S}^N \to \mathcal{N}$ we give a quantitative estimate of its rational homotopy group element ${\rm deg}([f]) \in \mathbb{R}$ in terms of its fractional Sobolev-norm or Hölder norm. That is, we show that for all $β\in (β_0({\rm deg}),1]$, \[ |{\rm deg}([f])|\leq C({\rm deg})\, [f]_{W^{β,\frac{N}β}(\mathbb{S}^N)}^{\frac{N+L({\rm deg})}β}, \] and \[ |{\rm deg}([f])|\leq C({\rm deg})\, [f]_{C^β(\mathbb{S}^N)}^{\frac{N+L({\rm deg})}β}. \] Here $C({\rm deg}) > 0$, $L({\rm deg}) \in \mathbb{N}$, $β_0({\rm deg}) \in (0,1)$ are computable from the rational homotopy group represented by ${\rm deg}$. This extends earlier work by Van Schaftingen and the second author on the Hopf degree to the Novikov's integral representation for rational homotopy groups as developed by Sullivan, Novikov, Hardt and Rivière.
In this paper, we formulate and prove a general compactness theorem for harmonic maps using Deligne-Mumford moduli space and families of curves. The main theorem shows that given a sequence of harmonic maps over a sequence of complex curves, there is a family of curves and a subsequence such that both the domains and the maps converge off the set of "non-regular" nodes. This provides a sufficient condition for a neck having zero energy and zero length. As a corollary, the following known fact can be proved: If all domains are diffeomorphic to $S^2$, both energy identity and zero distance bubbling hold.