Shige Peng, Detang Zhou
In this work we consider viscosity solutions to second order partial differential equations on Riemannian manifolds. We prove maximum principles for solutions to Dirichlet problem on a compact Riemannian manifold with boundary. Using a different method, we generalize maximum principles of Omori and Yau to a viscosity version.
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We prove comparison, uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations $F(x, u, du, d^{2}u)=0$ defined on a finite-dimensional Riemannian manifold $M$. Finest results (with hypothesis that require the function $F$ to be degenerate elliptic, that is nonincreasing in the second order derivative variable, and uniformly continuous with respect to the variable $x$) are obtained under the assumption that $M$ has nonnegative sectional curvature, while, if one additionally requires $F$ to depend on $d^{2}u$ in a unif…
We establish some perturbed minimization principles, and we develop a theory of subdifferential calculus, for functions defined on Riemannian manifolds. Then we apply these results to show existence and uniqueness of viscosity solutions to Hamilton-Jacobi equations defined on Riemannian manifolds.
Viscosity solutions are suitable notions in the study of nonlinear PDEs justified by estimates established via the maximum principle or the comparison principle. Here we prove that the isoperimetric profile functions of Riemannian manifolds with Ricci lower bound are viscosity super-solutions of some nonlinear differential equations. From these one can derive the isoperimetric inequalities of Lévy-Gromov and Bérard-Besson-Gallot, as well as some new comparison results.
A sub-Riemannian maximum modulus theorem
本文在卡诺群中证明了次黎曼最大模定理，并通过非平凡反例说明该定理关于右不变水平梯度的条件不可替换为左不变水平梯度，从而表明结果的最优性。
We investigate strong maximum (and minimum) principles for fully nonlinear second order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci's extremal operators, some singular operators like those modeled on the $p$- and $\infty$-Laplacian, and mean curvature type problems. As a byproduct, we establish new strong comparison principles for some second order uniformly elliptic problems when the manifold has nonnegative sectional curvature.
Maximum principles in unbounded Riemannian domains
该研究针对无界黎曼区域上带狄利克雷边界条件的二阶椭圆算子，结合环境流形与微分算子两类假设路径，开展最大值原理的证明研究，全文分为两个独立部分。
We give a definition of weak geodesics on prox-regular subsets of Riemannian manifolds as continuous curves with some weak regularities. Then obtaining a suitable Lipschitz constant of the projection map, we characterize weak geodesics on a prox-regular set with assigned end points as viscosity critical points of the energy functional.
If $U:[0,+\infty[\times M$ is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation $$\partial_tU+ H(x,\partial_xU)=0,$$ where $M$ is a not necessarily compact manifold, and $H$ is a Tonelli Hamiltonian, we prove the set $Σ(U)$, of points where $U$ is not differentiable, is locally contractible. Moreover, we study the homotopy type of $Σ(U)$. We also give an application to the singularities of a distance function to a closed subset of a complete Riemannian manifold.
In this paper we present a proof of a Neumann type maximum principle for the Laplace operator on compact Riemannian manifolds. A key p oint is the simple geometric nature of the constant in the a priori estimate of this maximum principle. In particular, this maximum principle can be applied to manifolds with Ricci curvature bounded from below and diameter bounded from above to yield a maximum estimate without dependence on a positive lower bound for the volume.
Resolving the viscosity operator ambiguity on Riemannian manifolds via a kinematic selection principle
本文研究了在黎曼流形上Navier-Stokes方程的不同粘性算子选择问题，通过运动构造原理唯一选择变形拉普拉斯算子，并证明了在特定边界条件下，变形拉普拉斯算子在超曲面上具有良好的 coercivity 性质，从而解决了分析上的障碍。
The strong maximum principle is proved to hold for weak (in the sense of support functions) sub- and super-solutions to a class of quasi-linear elliptic equations that includes the mean curvature equation for $C^0$ spacelike hypersurfaces in a Lorentzian manifold. As one application a Lorentzian warped product splitting theorem is given.
Based on works by Hopf, Weinberger, Hamilton and Evans, we state and prove the strong elliptic maximum principle for smooth sections in vector bundles over Riemannian manifolds and give some applications in Differential Geometry. Moreover, we use this maximum principle to obtain various rigidity theorems and Bernstein type theorems in higher codimension for minimal maps between Riemannian manifolds.
We establish a boundary maximum principle for free boundary minimal submanifolds in a Riemannian manifold with boundary, in any dimension and codimension. Our result holds more generally in the context of varifolds.
In this paper, we define and study sub-Riemannian structures on Banach manifolds. We obtain extensions of the Chow-Rashevski theorem for exact controllability, and give conditions for the existence of a Hamiltonian geodesic flow despite the lack of a Pontryagin Maximum Principle in the infinite dimensional setting.
Global propagation of singularities for magnetic mechanical systems
本文证明了闭Riemann流形上磁力学系统相关Hamilton-Jacobi方程粘性解的奇性全局传播，表明弱KAM解的奇点集在广义梯度流下保持不变，并提出了一种结合系统约化、重参数化流分析与正则化技术的几何证明方法。
Remarks on Hessian quotient equations on Riemannian manifolds
针对 Delanoë 和 Urbas 提出的相关问题，本文在二维黎曼流形上利用新测试函数与商算子凹性，证明实 Hessian 商方程具有无障碍二阶先验估计，并揭示其与复几何中 $J$-方程障碍的差异。
Multiplicity of solutions for Gross-Pitaevskii equations on Riemannian manifolds
本文研究了在黎曼流形上Gross-Pitaevskii方程解的多重性问题，通过临界点理论和Γ收敛方法，结合新的等周型问题结果，得出了解的多重性下界。
We consider the generalized evolution of compact level sets by functions of their normal vectors and second fundamental forms on a Riemannian manifold M. The level sets of a function $u:M\to\mathbb{R}$ evolve in such a way whenever u solves an equation $u_{t}+F(Du, D^{2}u)=0$, for some real function F satisfying a geometric condition. We show existence and uniqueness of viscosity solutions to this equation under the assumptions that M has nonnegative curvature, F is continuous off Du=0, (degenerate) elliptic, and locally invariant by parallel translation. We then prove that this approach is ge…
On the critical points of semi-stable solutions on convex domains of Riemannian surfaces
本文研究了常曲率二维模型空间凸域上半线性方程正半稳定解的临界点问题，通过辅助函数与拓扑度论证证明其恰有一个非退化临界点。
In this paper, we derive a new form of maximum principle for smooth functions on a complete noncompact Riemannian manifold $M$ for which there exists a bounded vector field $X$ such that $\langle\nabla f,X\rangle\geq 0$ on $M$ and $\mathrm{div} X\geq af$ outside a suitable compact subset} of $M$, for some constant $a>0$, under the assumption that $M$ has either polynomial or exponential volume growth. We then use it to obtain some straightforward applications to smooth functions and, more interestingly, to Bernstein-type results for hypersurfaces immersed into a Riemannian manifold endowed wit…
In this paper, by employ the cutoff function and the maximum principle, some Hamilton-Souplet-Zhang type gradient estimates for porous medium type equation are deduced. As a special case, an Hamilton-Souplet-Zhang type gradient estimates of the heat equation is derived which is different from the result of Souplet-Zhang. Furthermore, our results generalize those of Zhu. As application, some Livillous theorems for ancient solution are derived.
A boundary maximum principle for stationary pairs of varifolds with fixed contact angle
本文研究带光滑边界的紧 Riemannian 流形中满足固定接触角条件的平稳 varifold 对，并为其建立边界极大值原理。
In this paper we characterize the degenerate elliptic equations F(D^2u)=0 whose viscosity subsolutions, (F(D^2u) \geq 0), satisfy the strong maximum principle. We introduce an easily computed function f(t) for t > 0, determined by F, and we show that the strong maximum principle holds depending on whether the integral \int dy / f(y) near 0 is infinite or finite. This complements our previous work characterizing when the (ordinary) maximum principle holds. Along the way we characterize radial subsolutions.
The asymptotic behavior of the heat kernel of a Riemannian manifold gives rise to the classical concepts of parabolicity, stochastic completeness (or conservative property) and Feller property (or $C^{0}$-diffusion property). Both parabolicity and stochastic completeness have been the subject of a systematic study which led to discovering not only sharp geometric conditions for their validity but also an incredible rich family of tools, techniques and equivalent concepts ranging from maximum principles at infinity, function theoretic tests (Khas'minskii criterion), comparison techniques etc...…
We construct the first examples of continuous families of isospectral Riemannian metrics that are not locally isometric on closed manifolds, more precisely, on $S^n\times T^m$, where $T^m$ is a torus of dimension $m\ge 2$ and $S^n$ is a sphere of dimension $n\ge 4$. These metrics are not locally homogeneous; in particular, the scalar curvature of each metric is nonconstant. For some of the deformations, the maximum scalar curvature changes during the deformation.
The author discusses in some detail the old definitions of the curvature tensors for rigged metrized distributions on manifolds given by Schouten, Wagner, and Solov'ev. To calculate the Solov'ev sectional and Ricci curvatures for homogeneous sub-Riemannian manifolds, the author suggests to use in some cases special riggings of invariant completely non- holonomic distributions on manifolds. As a justification, we find a foliation on the cotangent bundle over a Lie group G whose leaves are tangent to invariant Hamiltonian vector fields for the Pontryagin-Hamilton func- tion. This function was ap…
In this paper we extend to non-compact Riemannian manifolds with boundary the use of two important tools in the geometric analysis of compact spaces, namely, the weak maximum principle for subharmonic functions and the integration by parts. The first one is a new form of the classical Ahlfors maximum principle whereas the second one is a version for manifolds with boundary of the so called Kelvin-Nevanlinna-Royden criterion of parabolicity. In fact, we will show that the validity of non-compact versions of these tools serve as a characterization of the Neumann parabolicity of the space. The mo…
Let $\,(M,g)\,$ be a $n$-dimensional Riemannian manifold and $\,Ω\,$ be any compact connected domain in $\,M$. We study the problem of finding the {\em maxima} of the functional $\, {\mathcal E} (Ω)\,$ (known as {\em torsional rigidity} associated to $Ω$) among all domains of prescribed volume $v$. Our results show that for a given Riemannian manifold which is strictly isoperimetric at one of its points the maximum of such functional is realized by the geodesic ball centered at this point. More generally, we prove estimates for the functional $\, {\mathcal E} (Ω)\,$ by comparison with symmetri…
Sub-Riemannian geodesics on the Heisenberg 3D nil-manifold
本文研究了3D Heisenberg群上次黎曼结构到Heisenberg 3D nil-manifold的投影，分析了测地流的动力学性质，并基于$G$中次黎曼球与距离的精确双边界估计了$M$中测地线的切割时间。
We prove that in Riemannian manifolds the $k$-th Steklov eigenvalue on a domain and the square root of the $k$-th Laplacian eigenvalue on its boundary can be mutually controlled in terms of the maximum principal curvature of the boundary under sectional curvature conditions. As an application, we derive a Weyl-type upper bound for Steklov eigenvalues. A Pohozaev-type identity for harmonic functions on the domain and the min-max variational characterization of both eigenvalues are important ingredients.
On the ground of origins of the theory of Lie groups and Lie algebras, their (co)adjoint representations, and the Pontryagin maximum principle for the time-optimal problem are given an independent foundation for methods of geodesic vector field to search for normal geodesics of left-invariant (sub-)Finsler metrics on Lie groups and to look for the corresponding locally optimal controls in (sub-)Riemannian case, as well as some their applications.
Frankel property and Maximum Principle at Infinity for complete minimal hypersurfaces
本文研究了4至7维黎曼流形中的完备极小超曲面，通过推广无穷远最大值原理并利用稳定极小超曲面分类，证明了四维流形中不相交极小超曲面界定等距乘积板的结论。
We generalize the Omori-Yau almost maximum principle of the Laplace-Beltrami operator on a complete Riemannian manifold $M$ to a second-order linear semi-elliptic operator $L$ with bounded coefficients and no zeroth order term. Using this result, we prove some Liouville-type theorems for a real-valued $C^{2}$ function $f$ on $M$ satisfying $L f \geq F(f)+ H(|\nabla f|) $ for real-valued continuous functions $F$ and $H$ on $\Bbb R$ such that $H(0)=0$.
Second Maximum of a Gaussian Random Field and Exact (t-)Spacing test
本文提出基于高斯随机场第二最大值的精确t-间距检验，用于检测稀疏替代项，具有高校准和功效。
Yau-type gradient estimates for p-harmonic functions on Riemannian manifolds with boundary under Dirichlet boundary condition
本文在带边黎曼流形上，利用边界径向障碍与内部最大值原理，将Yau型梯度估计推广至p-调和函数，并导出Liouville定理。
A special case of the main result states that a complete $1$-connected Riemannian manifold $(M^n,g)$ is isometric to one of the models $\mathbb R^n$, $S^n(c)$, $\mathbb H^n(-c)$ of constant curvature if and only if every $p\in M^n$ is a non-degenerate maximum of a germ of smooth functions whose Riemannian gradient is a conformal vector field.
Distortion estimates for barycentric coordinates on Riemannian simplices
本文研究了Riemannian简单形上barycentric坐标的失真估计，通过Karcher中心技术定义坐标图，并推导出坐标图的一阶和二阶导数估计，从而比较Riemannian度量与欧几里得度量的差异，证明误差随最大边长平方缩小。
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