Biharmonic maps into Sol and Nil spaces

p-双调和映射从梯度瑞奇溶子的利乌维尔定理

Liouville theorem on p-biharmonic map from gradient Ricci soliton

本文研究了从梯度瑞奇溶子出发的p-双调和映射，特别是二维香草溶子的性质。

四次热方程的古老解

Ancient Solutions to the Biharmonic Heat Equation

本文研究了四次热方程的古老解，并将其与多项式有界函数的空间维度联系起来，扩展了Colding和Minicozzi的研究。

Conformal change of Riemannian metrics and biharmonic maps

For the reduction ordinary differential equation due to Baird and Kamissoko \cite{BK} for biharmonic maps from a Riemannian manifold $(M^m,g)$ into another one $(N^n,h)$, we show that this ODE has no global positive solution for every $m\geq 5$. On the contrary, we show that there exist global positive solutions in the case $m=3$. As applications, for the the Riemannian product $(M^3,g)$ of the line and a Riemann surface, we construct the new metric $\widetilde{g}$ on $M^3$ conformal to $g$ such that every nontrivial product harmonic map from $M^3$ with respect to the original metric $g$ must …

Exemples de sous variétés de $\mathbb{S}^n$; les variétés proprement biharmoniques avec indice et les variétés proprement C--harmoniques

We call indexed-biharmonic maps, the solutions of a particular non linear elliptic PDE of order 4. This is a generalization of harmonic maps which verifies that biharmonic maps are biharmonic of index 0. The goal of this article is to study submanifolds of $\mathbb{S}^n$ whose inclusion is non harmonic and indexed-biharmonic (i.e. proper indexed-biharmonic). We give some pseudo-umbilical and non pseudo-umbilical examples and we study the link between the proper indexed-biharmonic submanifolds of $\mathbb{S}^n$ and the proper conformal-harmonic submanifolds of $\mathbb{S}^n$.

若干模型空间上的双调和函数与双本征函数

Biharmonic functions and bi-eigenfunctions on some model spaces

该研究针对球面上的双拉普拉斯算子给出简便公式与本征相关结果，推导了一类模型空间上双调和方程径向解的积分形式，证明其存在真双调和函数，给出空间形式上的显式例子，并完成穿孔欧氏空间上正拉普拉斯真双调和函数的分类。

Biharmonic wave maps into spheres

A global weak solution of the biharmonic wave map equation in the energy space for spherical targets is constructed. The equation is reformulated as a conservation law and solved by a suitable Ginzburg-Landau type approximation.

空间形式中的双调和超曲面

Biharmonic hypersurfaces in space forms

本文证明了非正截面曲率空间形式中任意维数的双调和超曲面均为极小，解决了双曲空间中广义陈省身猜想的超曲面情形；对单位球面中的双调和超曲面，推导出定量限制、逐点准则、数量曲率界及局部分类结果，为BMO猜想提供了证据。

双调和共形浸入到三维反德西特空间：刚性、局部存在性和抛物旋转族

Making Surfaces Biharmonic by Conformal Reparametrization in Anti-de Sitter Three-Space

研究非退化曲面到三维反德西特空间的双调和共形浸入，通过特定符号约定和相关量表示双调和方程，证明类空曲面相关性质，推导解析系统并证明局部存在性，给出零坐标重构及具体初始数据，记录类时抛物约化。

Rotationally symmetric biharmonic maps between models

The main aim of this paper is to study existence and stability properties of rotationally symmetric proper biharmonic maps between two $m$-dimensional models (in the sense of Greene and Wu). We obtain a complete classification of rotationally symmetric, proper biharmonic conformal diffeomorphisms in the special case that $m=4$ and the models have constant sectional curvature. Then, by introducing the Hamiltonian associated to this problem, we also obtain a complete description of conformal proper biharmonic solutions in the case that the domain model is ${\mathbb R}^4$. In the second part of t…

The biharmonic heat operator on edge manifolds and non-linear fourth order equations

We construct the biharmonic heat kernel for a suitable self-adjoint extension of the bi-Laplacian on a manifold with incomplete edge singularities. We employ a microlocal description of the biharmonic heat kernel to establish mapping properties of the corresponding biharmonic heat operator on certain Banach spaces. This yields short time existence for a class of semi-linear equations of fourth order, including for example the Cahn-Hilliard equation. We also obtain asymptotics of the solutions near the edge singularity.

Biharmonic wave maps: Local wellposedness in high regularity

We show the local wellposedness of biharmonic wave maps with initial data of sufficiently high Sobolev regularity and a blow-up criterion in the sup-norm of the gradient of the solutions. In contrast to the wave maps equation we use a vanishing viscosity argument and an appropriate parabolic regularization in order to obtain the existence result. The geometric nature of the equation is exploited to prove convergence of approximate solutions, uniqueness of the limit, and continuous dependence on initial data.

Quantization for biharmonic maps from non-collapsed degenerating Einstein 4-manifolds

For a sequence of extrinsic or intrinsic biharmonic maps $u_j: M_j\rightarrow N$ from a sequence of non-collapsed degenerating closed Einstein 4-manifolds $(M_j,g_j)$ with bounded Einstein constants, bounded diameters and bounded $L^2$ curvature energy into a compact Riemannian manifold $(N,h)$ with uniformly bounded biharmonic energy, we establish a compactness theory modular finitely many bubbles, which are finite energy biharmonic maps from $\mathbb{R}^4$, or from $\mathbb{R}^4 / Γ$ for some nontrivial finite group $Γ\subset SO(4)$, or from some complete, noncompact, Ricci flat, non-flat AL…