Three spheres theorem for p-harmonic functions

调和映照到球面在奇异或退化自由边界处的部分正则性

Partial Regularity for Harmonic Maps into Spheres at a Singular or Degenerate Free Boundary

本文证明了在域度量沿自由边界以$d^α$速率退化或奇异时，具有自由边界数据的弱平稳调和映照的部分正则性。

Lp-gradient harmonic maps into spheres and SO(N)

We consider critical points of the energy $E(v) := \int_{\mathbb{R}^n} |\nabla^s v|^{\frac{n}{s}}$, where $v$ maps locally into the sphere or $SO(N)$, and $\nabla^s = (\partial_1^s,\ldots,\partial_n^s)$ is the formal fractional gradient, i.e. $\partial_α^s$ is a composition of the fractional laplacian with the $α$-th Riesz transform. We show that critical points of this energy are Hölder continuous. As a special case, for $s = 1$, we obtain a new, more stable proof of Fuchs and Strzelecki's regularity result of $n$-harmonic maps into the sphere, which is interesting on its own.

Regularity of n/2-harmonic maps into spheres

We prove Hoelder continuity for n/2-harmonic maps from subsets of Rn into a sphere. This extends a recent one-dimensional result by F. Da Lio and T. Riviere to arbitrary dimensions. The proof relies on compensation effects which we quantify adapting an approach for Wente's inequality by L. Tartar, instead of Besov-space arguments which were used in the one-dimensional case. Moreover, fractional analogues of Hodge decomposition and higher order Poincare inequalities as well as several localization effects for nonlocal operators similar to the fractional laplacian are developed and applied.

Sphere-valued harmonic maps with surface energy and the $K_{13}$ problem

We consider an energy functional motivated by the celebrated $K_{13}$ problem in the Oseen-Frank theory of nematic liquid crystals. It is defined for sphere-valued functions and appears as the usual Dirichlet energy with an additional surface term. It is known that this energy is unbounded from below and our aim has been to study the local minimizers. We show that even having a critical point in a suitable energy space imposes severe restrictions on the boundary conditions. Having suitable boundary conditions makes the energy functional bounded and in this case we study the partial regularity …

球中自由边界 $(n+α)$-调和映射的弱能量恒等式

A weak energy identity for $(n+α)$-harmonic maps with a free boundary in a sphere

本文证明了球中自由边界 $(n+α)$-调和映射序列收敛到泡树，并建立了极限能量等于宏观极限能量与各泡能量加权和的弱能量恒等式。

On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres

This paper is devoted to the asymptotic analysis of a fractional version of the Ginzburg-Landau equation in bounded domains, where the Laplacian is replaced by an integro-differential operator related to the square root Laplacian as defined in Fourier space. In the singular Ginzburg-Landau limit, we show that solutions with uniformly bounded energy converge weakly to sphere valued 1/2-harmonic maps, i.e., the fractional analogues of the usual harmonic maps. In addition, the convergence holds in smooth functions spaces away from a (n-1)-rectifiable closed set of finite (n-1)-Hausdorff measure. …

Epsilon-regularity for p-harmonic maps at a free boundary on a sphere

We prove an $ε$-regularity theorem for vector-valued p-harmonic maps, which are critical with respect to a partially free boundary condition, namely that they map the boundary into a round sphere. This does not seem to follow from the reflection method that Scheven used for harmonic maps with free boundary (i.e., the case $p=2$): the reflected equation can be interpreted as a $p$-harmonic map equation into a manifold, but the regularity theory for such equations is only known for round targets. Instead, we follow the spirit of the last-named author's recent work on free boundary harmonic maps …

From refined estimates for spherical harmonics to a sharp multiplier theorem on the Grushin sphere

We prove a sharp multiplier theorem of Mihlin-Hörmander type for the Grushin operator on the unit sphere in $\mathbb{R}^3$, and a corresponding boundedness result for the associated Bochner-Riesz means. The proof hinges on precise pointwise bounds for spherical harmonics.

泛非唯一性最小调和映射从球体到球面

Generic non-uniqueness of minimizing harmonic maps from a ball to a sphere

本文研究了从三维球体到二维球面的最小调和映射的非唯一性，通过同调构造证明了通过小变化可使边界映射具有多个极小值。

到球面的稳定调和映射的最优正则性

Optimal regularity of stable harmonic maps to spheres

本文研究到球面的稳定调和映射的正则性，确定其奇异集余维数的下界，证明结果最优，并给出n小于k时n维到k维球面非常数调和映射的非平凡指标界。

The regularity of harmonic maps into spheres and applications to Bernstein problems

We show the regularity of, and derive a-priori estimates for (weakly) harmonic maps from a Riemannian manifold into a Euclidean sphere under the assumption that the image avoids some neighborhood of a half-equator. The proofs combine constructions of strictly convex functions and the regularity theory of quasi-linear elliptic systems. We apply these results to the spherical and Euclidean Bernstein problems for minimal hypersurfaces, obtaining new conditions under which compact minimal hypersurfaces in spheres or complete minimal hypersurfaces in Euclidean spaces are trivial.

Quaternionic spherical harmonics and a sharp multiplier theorem on quaternionic spheres

A sharp $L^p$ spectral multiplier theorem of Mihlin--Hörmander type is proved for a distinguished sub-Laplacian on quaternionic spheres. This is the first such result on compact sub-Riemannian manifolds where the horizontal space has corank greater than one. The proof hinges on the analysis of the quaternionic spherical harmonic decomposition, of which we present an elementary derivation.

低能ε-调和映射到圆球

Low energy $\varepsilon$-harmonic maps into the round sphere

研究低能ε-调和映射从正曲率曲面到圆球的分类，发现所有度数为±1的映射均接近气泡配置，气泡半径与ε^{1/4}成正比。

A gap theorem for $α$-harmonic maps between two-spheres

In this paper we consider approximations introduced by Sacks-Uhlenbeck of the harmonic energy for maps from $S^2$ into $S^2$. We continue the analysis in [6] about limits of $α$-harmonic maps with uniformly bounded energy. Using a recent energy identity in [7], we obtain an optimal gap theorem for the $α$-harmonic maps of degree $-1, 0$ or $1$.

三次球定理对于调和函数（非同心情况)

Three-spheres theorem for harmonic functions (non-concentric case)

研究非同心球体中调和函数的三次球定理，通过倒置技术扩展到相关非同心非接触球体，并应用于小量传播和调和函数的唯一性。

到受扰动球面的调和映射热流的稳定爆破

Stable blowup for the harmonic map heat flow into perturbed spheres

该研究针对3至6维、靶流形为单位球面微小扰动的调和映射热流，构造了稳定自相似爆破解并证明其渐近非线性稳定性，采用类似波映射方程相关问题的摄动方法完成论证。

Arithmetic three-spheres theorems for quasilinear Riccati type inequalities

We consider arithmetic three-spheres inequalities to solutions of certain second order quasilinear elliptic differential equations and inequalities with a Riccati-type drift term.

Stability of stationary equivariant wave maps from the hyperbolic plane

In this paper we initiate the study of equivariant wave maps from 2d hyperbolic space into rotationally symmetric surfaces. This problem exhibits markedly different phenomena than its Euclidean counterpart due to the exponential volume growth of concentric geodesic spheres on the domain. In particular, when the target is the 2-sphere, we find a family of equivariant harmonic maps indexed by a parameter that measures how far the image of each harmonic map wraps around the sphere. These maps have energies taking all values between zero and the energy of the unique co-rotational Euclidean harmoni…

Zero Comparison Theorem of Sturm-Liouville Type for Harmonic Heat Flow

In this paper, we will prove a zero comparison theorem of Sturm-Liouville type for linearized harmonic heat flow for rotational symmetric solutions.

从圆周到球面的热流到调和映射的全局能控性

Global controllability to harmonic maps of the heat flow from a circle to a sphere

本文研究从圆周到球面的调和映射热流的全局能控性与镇定问题，结合控制理论与微分几何等方法，证明了小时间局部能控性及同伦类内的全局能控性等重要性质。

On the size of the singular set of minimizing harmonic maps into the 2-sphere in dimension four and higher

We extend the results of our recent preprint [arXiv: 1811.00515] into higher dimensions $n \geq 4$. For minimizing harmonic maps $u\in W^{1,2}(Ω,\mathbb{S}^2)$ from $n$-dimensional domains into the two dimensional sphere we prove: (1) An extension of Almgren and Lieb's linear law, namely \[\mathcal{H}^{n-3}(\textrm{sing} u) \le C \int_{\partial Ω} |\nabla_T u|^{n-1} \,d\mathcal{H}^{n-1};\] (2) An extension of Hardt and Lin's stability theorem, namely that the size of singular set is stable under small perturbations in $W^{1,n-1}$ norm of the boundary.

球谐函数与球面上的点构型

Spherical harmonics and point configurations on the sphere

该研究构造了极点为间距合适的点构型的二维单位球面上的球谐函数，发现当极点构型均匀分布时球谐序列呈现量子遍历性，上确界范数受极点聚类程度定量控制。

球面约束与调和映射流：低模态强迫下的可控性与可达性

Sphere Constraints and Harmonic Map Flow: Controllability and Reachability by Low-Mode Forcing

研究退化强迫下球面约束演化方程的可控性与可达性，以调和映射热流为例。通过利用几何结构转化为控制仿射系统，分析李代数，证明迭代李括号可产生新方向，为控制流形值演化方程提供李代数框架。

低能 $α$-调和映射到球面

Low-energy $α$-harmonic maps into the round sphere

本文对常曲率闭非球面黎曼曲面到圆球的低能 $α$-调和映射按气泡尺度和中心分类，证明在能量接近临界值时度为一的映射在显式函数的临界点以特定尺度产生气泡。

Low energy levels of harmonic maps into analytic manifolds

We consider the energy spectrum $Ξ_E(N)$ of harmonic maps from the sphere into a closed Riemannian manifold $N$. While a well known conjecture asserts that $Ξ_E(N)$ is discrete whenever $N$ is analytic, for most analytic targets it is only known that any potential accumulation point of the energy spectrum must be given by the sum of the energies of at least two harmonic spheres. The lowest energy level that could hence potentially be an accumulation point of $Ξ_E$ is thus $2 E_{min}$. In the present paper we exclude this possibility for generic 3 manifolds and prove additional results that est…

受控调和映射热流及从圆盘到球面的爆破

Controlled harmonic map heat flow and blow-up from the disk to the sphere

本文研究圆盘到球面的单余旋调和映射热流，提出适应控制的粘合策略以阻止爆破，并证明边界控制可驱动爆破或阻止无限时间浓度。

Shrinkers, expanders, and the unique continuation beyond generic blowup in the heat flow for harmonic maps between spheres

Using mixed analytical and numerical methods we investigate the development of singularities in the heat flow for corotational harmonic maps from the $d$-dimensional sphere to itself for $3\leq d\leq 6$. By gluing together shrinking and expanding asymptotically self-similar solutions we construct global weak solutions which are smooth everywhere except for a sequence of times $T_1<T_2<...<T_k<\infty$ at which there occurs the type I blow-up at one of the poles of the sphere. We show that in the generic case the continuation beyond blow-up is unique, the topological degree of the map changes by…

调和球面的Ginzburg--Landau逼近中的刚性

Rigidity in the Ginzburg--Landau approximation of harmonic spheres

该研究针对调和球面的Ginzburg--Landau逼近问题，证明并非所有球面间调和映射都可作为Ginzburg--Landau临界点极限，一次可逼近调和映射仅为旋转，并通过刚性定理给出小参数下低能临界点的具体形式。

Computer-Assisted Proof of the Main Theorem of 'The Classification of Branched Willmore Spheres in the $3$-Sphere and the $4$-Sphere'

We provide a computer-assisted proof of the holomorphy of the quartic and the octic meromorphic differentials arising in the main Theorem 4.11 of our paper 'The Classification of Branched Willmore spheres in the $3$-Sphere and the $4$-Sphere' (arXiv:1706.01405), using the free mathematical software Sage.

临界维数中球面映射的能量恒等式

Energy identity for stationary biharmonic mappings into spheres in supercritical dimensions

本文在超临界维数n≥5中，通过改进Lin-Rivière的方法，建立了驻定双调和映射到球面的能量恒等式。

Vanishing theorem and finiteness theorem for $p$-harmonic $1$ form

In this paper, we will show vanishing theorem of $p$ harmonic $1$ form on submanifold $M$ in $ \bar{M} $ whose BiRic curvature satisfying $ \overline{\mathrm{BiRic}}^a \geq Φ_a(H,S) $. As an corollary, we can get the corresponding theorem for $ p $ harmonic function and $ p $ harmonic map. We also investigate the finiteness problem of $p$ harmonic $1$ form on submanifold $M$ in $ \bar{M} $ whose BiRic curvature satisfying $ \overline{\mathrm{BiRic}}^a \geq -k^2 $.

Fractional div-curl quantities and applications to nonlocal geometric equations

We investigate a fractional notion of gradient and divergence operator. We generalize the div-curl estimate by Coifman-Lions-Meyer-Semmes to fractional div-curl quantities, obtaining, in particular, a nonlocal version of Wente's lemma. We demonstrate how these quantities appear naturally in nonlocal geometric equations, which can be used to obtain a theory for fractional harmonic maps analogous to the local theory. Firstly, regarding fractional harmonic maps into spheres, we obtain a conservation law analogous to Shatah's conservation law and give a new regularity proof analogous to Hélein's f…

Sphere Theorems in Geometry

In this paper, we give a survey of various sphere theorems in geometry. These include the topological sphere theorem of Berger and Klingenberg as well as the differentiable version obtained by the authors. These theorems employ a variety of methods, including geodesic and minimal surface techniques as well as Hamilton's Ricci flow. We also obtain here new results concerning complete manifolds with pinched curvature.

一种生成调和映射的框架

A framework for producing harmonic maps

该研究提出从单位球到欧几里得球面生成调和映射族的框架，基于托特工作，为广义径向投影提供理论背景，还指出这些映射族可用于构建相关变分问题的解。

Stability of trace theorems on the sphere

We prove stable versions of trace theorems on the sphere in $L^2$ with optimal constants, thus obtaining rather precise information regarding near-extremisers. We also obtain stability for the trace theorem into $L^q$ for $q > 2$, by combining a refined Hardy-Littlewood-Sobolev inequality on the sphere with a duality-stability result proved very recently by Carlen. Finally, we extend a local version of Carlen's duality theorem to establish local stability of certain Strichartz estimates for the kinetic transport equation.