L. Pestov, G. Uhlmann
We prove that knowing the length of geodesics joining points on the boundary of a two-dimensional, compact, simple Riemannian manifold with boundary, we can determine uniquely the Riemannian metric up to the natural obstruction.
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Torsional Rigidity and Spherical Deficit for a Dirichlet Problem on Riemannian Manifolds
研究Ricci曲率下有界的黎曼流形上Dirichlet问题的扭转刚度不等式和球面亏损积分恒等式，并应用于Einstein流形。
Rigidity results for the $p$-Laplace type equations on compact Riemannian manifolds
本文利用非线性流和 carré du champ 方法，在紧致黎曼流形上证明了带指数非线性的 $p$-Laplace 型与 $n$-Laplace 方程的刚性结果，并导出了插值不等式。
Rigidity results for Serrin's overdetermined problems in Riemannian manifolds
本文研究Riemannian流形上的Serrin超定问题，通过共形变换与P函数方法证明Pohozoaev型恒等式及Serrin型刚性结果，并利用广义归一化壁面剪应力界限得出Dirichlet问题的对称性结果。
We prove two rigidity theorems for maps between Riemannian manifolds. First, we prove that a Lipschitz map $f:M\to N$ between two oriented Riemannian manifolds, whose differential is almost everywhere an orientation-preserving isometry, is an isometric immersion. This theorem was previously proved using regularity theory for conformal maps; we give a new, simple proof, by generalizing the Piola identity for the cofactor operator. Second, we prove that if there exists a sequence of mapping $f_n:M\to N$, whose differentials converge in $L^p$ to the set of orientation-preserving isometries, then …
Given a compact manifold with boundary with unknown Riemannian metric. The problem is to reconstruct the metric in a class of conformal metrics from knowledge of lengths of all closed geodesics (kinematic data). An integral inequality is stated which implies uniqueness and stability for this problem. If the conformal class is not known a unique reconstruction is not possible since of shortage of information. It is proved that the list of all geodesic lengths is sufficient for unique determination of a Riemannian metric in a compact surface with boundary up to an automorphism which fix the boun…
We study the boundary rigidity problem for compact Riemannian manifolds with boundary $(M,g)$: is the Riemannian metric $g$ uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function $ρ_g(x,y)$ known for all boundary points $x$ and $y$? We prove in this paper global uniqueness and stability for the boundary rigidity problem for generic simple metrics. More specifically, we show that there exists a generic set $\mathcal{G}$ of simple Riemannian metrics and an open dense set $\mathcal{U}\subset \mathcal{G}\times\mathcal{G}$, such that any two Riemannian …
The Liouville equation on Riemannian surfaces: the role of volume growth in classification and rigidity results
研究具有非负里奇曲率的黎曼曲面上的刘维尔方程，通过假设解的渐近下界，建立解与环境流形的分类结果，揭示流形体积增长与分类的紧密联系。
Given a Riemannian spin^c manifold whose boundary is endowed with a Riemannian flow, we show that any solution of the basic Dirac equation satisfies an integral inequality depending on geometric quantities, such as the mean curvature and the O'Neill tensor. We then characterize the equality case of the inequality when the ambient manifold is a domain of a Kähler-Einstein manifold or a Riemannian product of a Kähler-Einstein manifold with R (or with the circle S^1).
We study a new class of rank two sub-Riemannian manifolds encompassing Riemannian manifolds, CR manifolds with vanishing Webster-Tanaka torsion, orthonormal bundles over Riemannian manifolds, and graded nilpotent Lie groups of step two. These manifolds admit a canonical horizontal connection and a canonical sub-Laplacian. We construct on these manifolds an analogue of the Riemannian Ricci tensor and prove Bochner type formulas for the sub-Laplacian. As a consequence, it is possible to formulate on these spaces a sub-Riemannian analogue of the so-called curvature dimension inequality. Sub-Riema…
We obtain upper bounds on the heat content and on the torsional rigidity of a complete Riemannian manifold M, assuming a generalized Hardy inequality for the Dirichlet Laplacian on M.
We prove that polyharmonic maps of arbitrary order from complete nonparabolic Riemannian manifolds to arbitrary Riemannian manifolds must be harmonic if certain smallness and integrability conditions hold.
We formulate stochastic partial differential equations on Riemannian manifolds, moving surfaces, general evolving Riemannian manifolds (with appropriate assumptions) and Riemannian manifolds with random metrics, in the variational setting of the analysis to stochastic partial differential equations. Considering mainly linear stochastic partial differential equations, we establish various existence and uniqueness theorems.
We are concerned with the global weak rigidity of the Gauss-Codazzi-Ricci (GCR) equations on Riemannian manifolds and the corresponding isometric immersions of Riemannian manifolds into the Euclidean spaces. We develop a unified intrinsic approach to establish the global weak rigidity of both the GCR equations and isometric immersions of the Riemannian manifolds, independent of the local coordinates, and provide further insights of the previous local results and arguments. The critical case has also been analyzed. To achieve this, we first reformulate the GCR equations with div-curl structure …
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.
In this short note, we find a new gap phenomena on Riemannian manifolds, which says that for any complete noncompact Riemannian manifold with nonnegative Ricci curvature, if the scalar curvature decays faster than quadratically, then it is Ricci flat.
Let $σ$ be the scattering relation on a compact Riemannian manifold $M$ with non-necessarily convex boundary, that maps initial points of geodesic rays on the boundary and initial directions to the outgoing point on the boundary and the outgoing direction. Let $\ell$ be the length of that geodesic ray. We study the question of whether the metric $g$ is uniquely determined, up to an isometry, by knowledge of $σ$ and $\ell$ restricted on some subset $D$. We allow possible conjugate points but we assume that the conormal bundle of the geodesics issued from $D$ covers $T^*M$; and that those geodes…
We derive the existence of $C^{1,1}$-solutions to the Dirichlet problem for degenerate fully nonlinear elliptic equations on Riemannian manifolds under appropriate assumptions.
We prove a non-vanishing result for the $L_{q,p}$-cohomology of complete simply-connected Riemannian manifolds with pinched negative curvature.
We study some qualitative properties of ancient solutions of superlinear heat equations on a Riemannian manifold, with particular interest in positivity and constancy in space.
On a Local Radial Rigidity Phenomenon of Elliptic Systems on Riemannian Manifolds
本文通过将一般阶椭圆系统约化为欧拉型奇异常微分方程，研究其局部径向刚性现象，并应用于共形几何中的Yamabe和Paneitz型预定曲率问题、调和映射的高阶类似问题以及Kähler流形上的Calabi型问题。
Symmetry and rigidity results for Serrin's overdetermined type problems in weighted Riemannian manifolds
研究带权流形上有界域上的Serrin过定边界值问题，建立域闭合为紧致时的刚性结果，分析解与周围流形几何关系，并探讨非紧闭合域在欧几里得空间形变下的情况。
In this note we show how results in \cite{BaudoinBonnefont2016, BaudoinGarofalo2013, CoulhonJiangKoskelaSikora2017} yield the Cheng-Yau estimate on two classes of sub-Riemannian manifolds: Carnot groups and sub-Riemannian manifolds satisfying a generalized curvature-dimension inequality.
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.
Classification results, rigidity theorems and semilinear PDEs on Riemannian manifolds: a P-function approach
该研究采用P函数方法，分析完备非紧黎曼流形上半线性椭圆方程解的分类与流形刚性，得到临界方程正解分类、流形刚性等结果，部分结论在欧氏情形下也具新颖性。
We study the existence of extremal functions on compact Riemannian manifold wich is locally Euclidean .
This paper deals with bounded solutions of quasilinear elliptic equations on Riemannian manifolds satisfying special condition.
In this short note we review some facts about elliptic differential operators on Riemannian manifolds.
On compact Riemannian manifolds, we prove a decomposition theorem for arbitrarily bounded energy sequence of solutions of a singular elliptic equation.
We propose a global scheme for a controlled Navier-Stokes equation system on compact smooth Riemannian manifolds.
We give some estimates of type sup $\times$ inf on Riemannian manifold of dimension 5.
We show the existence of nodal solutions to perturbed quasilinear elliptic equations with critical Sobolev exponent on compact Riemannian manifolds. A nonexistence result is also given.
We give an estimate of type sup $\times$ inf on Riemannian manifold of dimension 4 for a Yamabe type equation.
The relation between the number of solutions of a nonlinear equation on a Riemannian manifold and the topology of the manifold itself is studied. The technique is based on Ljusternik-Schnirelmann category and Morse theory.
We find new polynomial upper bounds for the size of nodal sets of eigenfunctions when the Riemannian manifold has a Gevrey or quasianalytic regularity.
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