Jian Zhai
We summarize the proofs for the s-injectivity of the tensor tomography problem on compact Riemannian manifolds with boundaries in [Dairbekov, Inverse Problems, 22: 431, 2006] and [Paternain-Salo-Uhlmann, Math. Ann., 363: 305-362, 2015] under certain geometric assumptions.
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In this paper we consider the Gaussian thermostat ray transform on both closed Riemannian surfaces and compact Riemannian surfaces with boundary. We establish certain results on the injectivity of the thermostat ray transform and the surjectivity of its adjoint.
This PhD dissertation is concerned with integral geometric inverse problems. The geodesic ray transform is an operator that encodes the line integrals of a function along geodesics. The dissertation establishes many conditions when such information determines a function uniquely and stably. A new numerical model for computed tomography imaging is created as a part of the dissertation. The introductory part of the dissertation contains an introduction to inverse problems and mathematical models associated to computed tomography. The main focus is in definitions of integral geometry problems, su…
We show that a complete Riemannian manifold with boundary is uniquely determined, up to an isometry, by its distance difference representation on the boundary. Unlike previously known results, we do not impose any restrictions on the boundary.
We survey some results on travel time tomography. The question is whether we can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as geometry problems, the boundary rigidity problem and the lens rigidity problem. The boundary rigidity problem is whether we can determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points. The lens rigidity problem problem is to determine a Riemannian metric of a Riemannian manifold with boundary …
Let $(M,g)$ be a complete three dimensional Riemannian manifold with boundary $\partial M$. Given smooth functions $K(x)>0$ and $c(x)$ defined on $M$ and $\partial M$, respectively, it is natural to ask whether there exist metrics conformal to $g$ so that under these new metrics, $K$ is the scalar curvature and $c$ is the boundary mean curvature. All such metrics can be described by a prescribing curvature equation with a boundary condition. With suitable assumptions on $K$,$c$ and $(M,g)$ we show that all the solutions of the equation can only blow up at finite points over each compact subset…
In this paper we consider a set $E\subsetΩ$ with prescribed mean curvature $f\in C(Ω)$ and Euclidean Lipschitz boundary $\partial E=Σ$ inside a three-dimensional contact sub-Riemannian manifold $M$. We prove that if $Σ$ is locally a regular intrinsic graph, the characteristic curves are of class $C^2$. The result is shape and improves the ones contained in \cite{MR2583494} and \cite{GalRit15}.
We obtain existence of double bubbles of large and constant mean curvatures in Riemannian manifolds. These arise as perturbations of geodesic standard double bubbles centered at critical points of the ambient scalar curvature and aligned along eigen-vectors of the ambient Ricci tensor. We also obtain general multiplicity results via Lusternik-Schnirelman theory, and extra ones in case of double bubbles whose opposite boundaries have the same mean curvature.
Pestov identities and X-ray tomography on manifolds of low regularity
本文证明了在 $C^{1,1}$ 正则性的简单 Riemannian 流形上测地 X 射线变换的逆性，重新定义了粗糙几何下的简单性，并在单位球丛上建立了微分与曲率算子演算。
Tomography of 1-forms on a gas giant
本文研究了气态巨行星流形上1形式的测地X射线变换的单射性，通过Pestov恒等式和渐进行为分析证明其在适当光滑结构下具有单射性质。
Tensor tomography on negatively curved manifolds of low regularity
本文研究了低正则性负曲率流形上的张量层析成像问题，证明了 $C^{1,1}$ 度量简单 Riemannian 流形上测地 X 射线变换的螺线管式内射性，核心方法是利用非光滑单位球丛上输运方程的 Pestov 能量估计并对光滑证明方法进行实质性修改。
We prove positive mass theorems for asymptotically hyperbolic and asymptotically locally hyperbolic Riemannian manifolds with black-hole-type boundaries.
Recently, the old notion of causal boundary for a spacetime V has been redefined in a consistent way. The computation of this boundary $\partial V$ for a standard conformally stationary spacetime V = R x M, suggests a natural compactification $M_B$ associated to any Riemannian metric on M or, more generally, to any Finslerian one. The corresponding boundary $\partial_B M$ is constructed in terms of Busemann-type functions. Roughly, $\partial_B M$ represents the set of all the directions in M including both, asymptotic and "finite" (or "incomplete") directions. This Busemann boundary $\partial_…
Scalar curvature deformations with non-compact boundaries
本文为具有非紧边界的光滑流形开发了保持标量曲率下界的黎曼度量族形变原理，用于加强边界条件，并证明了具有正标量曲率与平均凸边界的度量不存在性结果。
This is the introductory part of my PhD thesis on inverse problems arising in medical and seismic imaging. The topics include X-ray tomography of scalar and vector fields with partial data, higher order fractional Calderón problems, travel time tomography on Riemannian and Finsler manifolds, and unique continuation of fractional Laplacians. The thesis additionally includes these articles: arXiv:1909.05585, arXiv:2001.06210, arXiv:2006.05790, arXiv:2008.10227, arXiv:2009.01043, arXiv:2010.11484, arXiv:2103.14385.
On a compact Riemannian manifold with boundary, the absolute and relative cohomology groups appear as certain subspaces of harmonic forms. DeTurck and Gluck showed that these concrete realizations of the cohomology groups decompose into orthogonal subspaces corresponding to cohomology coming from the interior and boundary of the manifold. The principal angles between these interior subspaces are all acute and are called Poincare duality angles. This paper determines the Poincare duality angles of a collection of interesting manifolds with boundary derived from complex projective spaces and fro…
A Riemannian Extension of a Quadrature Surface Free Boundary Problem: Stability and Optimality
该研究将求积曲面自由边界问题从欧氏环境扩展至紧黎曼流形，建立其紧性、狄利克雷问题稳定性、一阶最优性条件及相关比较原理，为该框架的黎曼扩展提供了严格依据。
We prove that any smooth Riemannian manifold of non-negative scalar curvature and with a strictly mean convex and compact boundary component can be (C^2) extended beyond the component to have non-negative scalar curvature and to enjoy anyone of the following three types of (new) boundary: strictly convex, totally geodesic or strictly concave. The extension procedure can be applied for instance to "positive mass" type of theorems.
Is it possible to obtain unbounded minimal surfaces in certain asymptotically flat 3-manifolds as a limit of solutions to a natural mountain pass problem with diverging boundaries? In this work, we give evidence that this might be true by analyzing related aspects in the case of the exact Riemannian Schwarzschild manifold. More precisely, we observe that the simplest minimal surface in this space has Morse index one. We prove also a relationship between the length of the boundary and the density at infinity of general minimal surfaces satisfying a free-boundary condition along the horizon.
We initiate the study of X-ray tomography on sub-Riemannian manifolds, for which the Heisenberg group exhibits the simplest nontrivial example. With the language of the group Fourier Transform, we prove an operator-valued incarnation of the Fourier Slice Theorem, and apply this new tool to show that a sufficiently regular function on the Heisenberg group is determined by its line integrals over sub-Riemannian geodesics. We also consider the family of taming metrics $g_ε$ approximating the sub-Riemannian metric, and show that the associated X-ray transform is injective for all $ε>0$. This resul…
In this paper, we establish a min-max theory for constructing minimal disks with free boundary in any closed Riemannian manifold. The main result is an effective version of the partial Morse theory for minimal disks with free boundary established by Fraser. Our theory also includes as a special case the min-max theory for Plateau problem of minimal disks, which can be used to generalize the famous work by Morse-Thompkins and Shiffman on minimal surfaces in $\mathbf{R}^n$ to the Riemannian setting. More precisely, we generalize the min-max construction of minimal surfaces using harmonic replace…
In this paper we analyze the problem of the geodesic connectedness of subsets of Riemannian manifolds. By using variational methods, the geodesic connectedness of open domains (whose boundaries can be not differentiable and not convex) of a smooth Riemannian manifold is proved. In some cases also the convexity of the domain is obtained. Moreover we present examples of the applicability and of the independence of the assumptions. Finally we give an application to the existence of trajectories with fixed energy of dynamical systems.
In this article, we give results of prescribing scalar and mean curvature functions for metrics either pointwise conformal or conformally equivalent to a Riemannian metric that is equipped on a compact manifold with boundary, with dimensions at least $ 3 $. The results are classified by the sign of the first eigenvalue of the conformal Laplacian. This leads to a "Trichotomy Theorem" in terms of both scalar and mean curvature functions, which is a full extension of the "Trichotomy Theorem" given by Kazdan and Warner. We also discuss prescribing Gauss and geodesic curvature problems on compact R…
Tensor tomography and frame flow ergodicity for magnetic flows in higher dimensions
本文将测地流理论中的两个结果扩展到任意维流形上的磁流，研究磁射线变换并建立张量断层成像结果，分析磁框架流在紧缩条件下的遍历性，发展了适应磁流的黎曼几何概念。
We consider nonnegative solutions of the porous medium equation (PME) on a Cartan-Hadamard manifold whose negative curvature can be unbounded. We take compactly supported initial data because we are also interested in free boundaries. We classify the geometrical cases we study into quasi-hyperbolic, quasi-Euclidean and critical cases, depending on the growth rate of the curvature at infinity. We prove sharp upper and lower bounds on the long-time behaviour of the solutions in terms of corresponding bounds on the curvature. In particular we obtain a sharp form of the smoothing effect on such ma…
A Riemannian approach for PDE-constrained shape optimization over the diffeomorphism group using outer metrics
本文提出了一种基于外度量的黎曼方法，用于解决受偏微分方程约束的形状优化问题，通过分析微分同胚群的结构，结合欧拉半导数与经典形状导数的关系，优化电阻抗断层成像和二维桥梁设计。
In this note, we estimate the upper bound of volume of closed positively or nonnegatively curved Alexandrov space $X$ with strictly convex boundary. We also discuss the equality case. In particular, the Boundary Conjecture holds when the volume upper bound is achieved. Our theorem also can be applied to Riemannian manifolds with non-smooth boundary, which generalizes Heintze and Karcher's classical volume comparison theorem. Our main tool is the gradient flow of semi-concave functions.
Scattering rigidity of a Riemannian manifold allows one to tell the metric of a manifold with boundary by looking at the directions of geodesics at the boundary. Lens rigidity allows one to tell the metric of a manifold with boundary from the same information plus the length of geodesics. There are a variety of results about lens rigidity but very little is known for scattering rigidity. We will discuss the subtle difference between these two types of rigidities and prove that they are equivalent for two-dimensional simple manifolds with boundaries. In particular, this implies that two-dimensi…
Consider a Riemannian manifold in dimension $n\geq 3$ with strictly convex boundary. We prove the local invertibility, up to potential fields, of the geodesic ray transform on tensor fields of rank four near a boundary point. This problem is closely related with elastic \textit{qP}-wave tomography. Under the condition that the manifold can be foliated with a continuous family of strictly convex hypersurfaces, the local invertibility implies a global result. One can straightforwardedly adapt the proof to show similar results for tensor fields of arbitrary rank.
A blow-up method to prescribed mean curvature graphs with fixed boundaries
本文运用Schoen和Yau的爆破方法，结合曲率估计与Nc-f条件，解决了黎曼流形上预定平均曲率Dirichlet问题，并应用于PMC Plateau问题。
We prove the validity of an inequality involving a mean of the area and the length of the boundary of immersed disks whose boundaries are homotopically non-trivial curves in an oriented compact manifold which possesses convex mean curvature boundary, positive escalar curvature and admits a map to $\mathbb{D}^2\times T^{n}$ with nonzero degree, where $\mathbb{D}^2$ is a disk and $T^n$ is an $n$-dimensional torus. We also prove a rigidity result for the equality case when the boundary is totally geodesic. This can be viewed as a partial generalization of a result due to Lucas Ambrózio in \cite{A…
We study boundary value problems for the Dirac operator on Riemannian Spin$^c$ manifolds of bounded geometry and with noncompact boundary. This generalizes a part of the theory of boundary value problems by C. Bär and W. Ballmann for complete manifolds with closed boundary. As an application, we derive the lower bound of Hijazi-Montiel-Zhang, involving the mean curvature of the boundary, for the spectrum of the Dirac operator on the noncompact boundary of a Spin$^c$ manifold. The limiting case is then studied and examples are then given.
We prove the non-vanishing of the CMC flux of the boundaries of certain Riemannian manifolds with constant mean curvature.
We study a family of gradient obstacle problems on a compact Riemannian manifold. We prove that the solutions of these free boundary problems are uniformly semiconcave and, as a consequence, we obtain some fine convergence results for the solutions and their free boundaries. Precisely, we show that the elastic and the $λ$-elastic sets of the solutions Hausdorff converge to the cut locus and the $λ$-cut locus of the manifold.
Given two compact Riemannian manifolds with boundary $M_1$ and $M_2$ such that their respective boundaries $Σ_1$ and $Σ_2$ admit neighborhoods $Ω_1$ and $Ω_2$ which are isometric, we prove the existence of a constant $C$, which depends only on the geometry of $Ω_1\congΩ_2$, such that $|σ_k(M_1)-σ_k(M_2)|\leq C$ for each $k\in\mathbb{N}$. This follows from a quantitative relationship between the Steklov eigenvalues $σ_k$ of a compact Riemannian manifold $M$ and the eigenvalues $λ_k$ of the Laplacian on its boundary. Our main result states that the difference $|σ_k-\sqrt{λ_k}|$ is bounded above …
This article is the sequel to our previous paper [LS] dealing with the near-equality case of the Positive Mass Theorem. We study the near-equality case of the Penrose Inequality for the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature whose boundaries are outermost minimal hypersurfaces. Specifically, we prove that if the Penrose Inequality is sufficiently close to being an equality on one of these manifolds, then it must be close to a Schwarzschild space with an appended cylinder, in the sense of Lipschitz Distance. Since the …
Mass, Staticity, and a Riemannian Penrose Inequality for Weighted Manifolds
本文基于加权曲率映射推导出加权质量，并定义加权质心，通过共形关系证明静态度量与加权静态度量等价，进而推导出加权流形上的黎曼Penrose不等式。
Quantitative tightness for three-dimensional contact manifolds: a sub-Riemannian approach
本文采用次黎曼方法研究三维切触流形的定量紧性，引入切触Jacobi曲线给出紧性半径下界，验证经典模型下估计的紧性，还得到K-切触情形下Cartan-Hadamard定理的切触类比。
Let $\overline{M}$ be a smooth manifold with boundary $\partial M$ and interior $M$. Consider an affine connection $\nabla$ on $M$ for which the boundary is at infinity. Then $\nabla$ is projectively compact of order $α$ if the projective structure defined by $\nabla$ smoothly extends to all of $\overline{M}$ in a specific way that depends on no particular choice of boundary defining function. Via the Levi--Civita connection, this concept applies to pseudo--Riemannian metrics on $M$. We study the relation between interior geometry and the possibilities for compactification, and then develop th…
Structure theory of $BV$ functions and finite perimeter sets on Riemannian manifolds
研究黎曼流形上有界变差函数与有限周长集的结构理论，通过构建局部化框架融合相关技术，恢复欧几里得理论关键结果，为边值问题提供背景，证明有限周长集逼近结果及应用于毛细问题能量泛函Gamma收敛。
We study the stability of the Positive Mass Theorem using the Intrinsic Flat Distance. In particular we consider the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces whose boundaries are either outermost minimal hypersurfaces or are empty. We prove that a sequence of these manifolds whose ADM masses converge to zero must converge to Euclidean space in the pointed Intrinsic Flat sense. In fact we provide explicit bounds on the Intrinsic Flat Distance between annular regions in the manifold…
Limits of manifolds with boundary II
本文研究满足曲率下界和直径上界的带边紧黎曼流形的Gromov-Hausdorff收敛与坍塌，描述极限空间的局部几何结构，并建立Lipschitz同伦稳定性。
Isoperimetric Regions in Anisotropically Scaled Product Manifolds
该研究探讨了在各向异性缩放的乘积流形中，等周区域的结构特性，证明了在特定条件下，等周区域可分解为两个流形的乘积。
This is a slightly altered version of the authors thesis from 2014. In the first main part we show that the quotient space of a compact, simply connected and nonnegatively curved Riemannian 4-manifold by an effective, isometric circle-action admits an approximation in Gromov-Hausdorff topology by smooth, positively curved Riemannian 3-manifolds. In the second main part we show that a compact, nonnegatively curved and fixed point homogeneous manifold is diffeomorphic to the unit normal bundles of two smooth closed submanifolds glued together along their boundaries. As a corollary we show that a…
We study a Riemannian manifold equipped with a density which satisfies the Bakry--Émery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalized dimension). We first obtain a Poincaré-type inequality on its boundary assuming that the latter is locally-convex; this generalizes a purely Euclidean inequality of Colesanti, originally derived as an infinitesimal form of the Brunn-Minkowski inequality, thereby precluding any extensions beyond the Euclidean setting. A dual version for generalized mean-convex boundaries is also obt…
Let $(X, g_0)$ be a complete, simply connected Riemannian manifold with sectional curvatures $K_{g_0}$ satisfying $-b^2 \leq K_{g_0} \leq -1$ for some $b \geq 1$. Let $g_1$ be a Riemannian metric on $X$ such that $g_1 = g_0$ outside a compact in $X$, and with sectional curvatures $K_{g_1}$ satisfying $K_{g_1} \leq -1$. The identity map $id : (X, g_0) \to (X, g_1)$ is bi-Lipschitz, and hence induces a homeomorphism between the boundaries at infinity of $(X, g_0)$ and $(X, g_1)$, which we denote by $\hat{id}_{g_0, g_1} : \partial_{g_0} X \to \partial_{g_1} X$. We show that if the boundary map $\…
Metric Lie Groups. Carnot-Carathéodory spaces from the homogeneous viewpoint
本书探讨 Lie 群上左不变距离定义的非光滑几何，重点阐述 Carnot 群等度量 Lie 群在幂零群渐近锥、对称空间抛物边界及流形极限等数学语境中的核心作用。
We study Riemannian manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound. In our weighted setting, we prove several rigidity theorems for such manifolds with boundary. We conclude a rigidity theorem for the inscribed radii, a volume growth rigidity theorem for the metric neighborhoods of the boundaries, and various splitting theorems. We also obtain rigidity results for the smallest Dirichlet eigenvalues for the weighted p-Laplacians.
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