Jeff Viaclovsky
We prove estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds. These equations are not arbitrary, but arise naturally in the study of conformal geometry.
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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 derive gradient and second order {\em a priori} estimates for solutions of the Neumann problem for a general class of fully nonlinear elliptic equations on compact Riemannian manifolds with boundary. These estimates yield regularity and existence results.
In this note we establish the large time non-negativity of the heat kernel for a class of elliptic differential operators on closed, Riemannian manifolds, and apply this result to a problem from conformal differential geometry.
In this paper we are concerned with a class of elliptic differential inequalities with a potential both on $\erre^m$ and on Riemannian manifolds. In particular, we investigate the effect of the geometry of the underlying manifold and of the behavior of the potential at infinity on nonexistence of nonnegative solutions.
In this paper, using blow-up analysis, we prove a quantization result for an elliptic equation with critical exponential growth on compact Riemannian surface without boundary. Similar results for Euclidean space were obtained by Adimurthi-Struwe \cite{Adi-Stru}, Druet \cite{Druet}, Lamm-Robert-Struwe \cite{L-R-S}, Martinazzi \cite{Mart}, Martinazzi-Struwe \cite{Mar-Stru}, and Struwe \cite{Struwe} respectively.
In this paper we study the asymptotic behavior of second-order uniformly elliptic operators on weighted Riemannian manifolds. They naturally emerge when studying spectral properties of the Laplace-Beltrami operator on families of manifolds with rapidly oscillating metrics. We appeal to the notion of H-convergence introduced by Murat and Tartar. In our main result we establish an H-compactness result that applies to elliptic operators with measurable, uniformly elliptic coefficients on weighted Riemannian manifolds. We further discuss the special case of ``locally periodic'' coefficients and st…
We solve the Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds under essentially optimal structure conditions, especially with no restrictions to the curvature of the underlying manifold and the second fundamental form of its boundary. The main result (Theorem 1.1) includes a new (and optimal) result in the Euclidean case. We introduce some new ideas and methods in deriving a priori estimates, which can be used to treat other types of fully nonlinear elliptic and parabolic equations on real or complex manifolds.
This article studies the Dirichlet problem for a class of degenerate fully nonlinear elliptic equations on Riemannian manifolds with \textit{mean concave} boundary in the sense that the mean curvature of the boundary is \textit{nonpositive}. The proof is primarily based on a quantitative boundary estimate. Also, we obtain analogous results in complex variables. In Appendix, the subsolutions are also constructed on certain topologically product manifolds.
Multiplicity of solutions to a nonlinear elliptic problem on a Riemannian orbifold
本文利用摄影方法，依据最大局部群点之子流形的 Lusternik–Schnirelmann 畴数，建立了 Riemannian orbifold 上非线性椭圆问题解数的下界。
We derive a priori second order estimates for solutions of a class of fully nonlinear elliptic equations on Riemannian manifolds under some very general structure conditions. We treat both equations on closed manifolds, and the Dirichlet problem on manifolds with boundary without any geometric restrictions to the boundary except being smooth and compact. As applications of these estimates we obtain results on regularity and existence.
We investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in $M\times \mathbb R_+$, where $M$ is a complete noncompact Riemannian manifold. Under specific assumptions, we establish existence of solutions satisfying prescribed conditions at infinity, depending on the direction along which infinity is approached. Moreover, the large-time behavior of such solutions is studied. We consider also elliptic equations on $M$ with similar conditions at infinity.
We prove some inequalities of Payne-Pólya-Weinberger-Yang type for eigenvalues of fourth-order elliptic operators in weighted divergence form on complete Riemannian manifolds which generalizes the corresponding result for the clamped plate problem. We also prove estimates for lower order eigenvalues that contain some of the estimates from the literature. As an application of our results, we obtain eigenvalues estimates for the bi-drifted Cheng-Yau operator.
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…
Overdetermined elliptic problems on model Riemannian manifolds
本文在模型黎曼流形的超定椭圆问题背景下，证明了满足常数Dirichlet和Neumann边界条件的环形扇形域必为球面扇形且解具径向对称性，将经典刚性现象推广至更一般的框架。
We present a general existence proof for a wide class of non-linear elliptic equations which can be applied to problems with barrier conditions without specifying any assumptions guaranteeing the uniqueness or local uniqueness of particular solutions. As an application we prove the existence of closed hypersurfaces with curvature prescribed in the tangent bundle of an ambient Riemannian manifold $N$ without supposing any sign condition on the sectional curvatures $K_N$. A curvature flow wouldn't work in this situation, neither the method of successive approximation.
We consider least energy solutions to the nonlinear equation $-Δ_g u=f(r,u)$ posed on a class of Riemannian models $(M,g)$ of dimension $n\ge 2$ which include the classical hyperbolic space $\mathbb H^n$ as well as manifolds with unbounded sectional geometry. Partial symmetry and existence of least energy solutions is proved for quite general nonlinearities $f(r,u)$, where $r$ denotes the geodesic distance from the pole of $M$.
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.
On Polynomial Complex Hessian Equations I: Non-Linear Non-Spectral Elliptic Hessian Equations on Riemannian Manifolds
本文研究紧致黎曼流形上的一般完全非线性椭圆Hessian方程，通过弱化标准对称性假设，在推导先验估计后利用连续性方法求解。
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 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.
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.
Let $σ>1$ and let $M$ be a complete Riemannian manifold. In a recent work [9], Grigor$^{\prime}$yan and Sun proved that a pointwise upper bound of volume growth is sufficient for uniqueness of nonnegative solutions of elliptic inequality $$(*)\quad\qquad\qquad\qquad Δu(x)+u^σ(x)\leq 0,\qquad x\in M.\quad\qquad \qquad\qquad $$ In this note, we improve their result to that an \emph{integral condition} on volume growth implies the same uniqueness of ($*$). It is inspired by the well-known Varopoulos-Grigor$^{\prime}$yan's criterion for parabolicity of $M$.
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.
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.
We develop some of the basic theory for the obstacle problem on Riemannian Manifolds, and we use it to establish a mean value theorem. Our mean value theorem works for a very wide class of Riemannian manifolds and has no weights at all within the integral.
We prove the existence of multiple closed geodesics on non-compact cylindrica manifolds.
In this article we establish a local parabolic almost monotonicity formula for two phase free boundary problems on Riemannian manifolds, which is an extension of a work of Edquist-Petrosyan.
We prove a Duistermaat-Guillemin trace formula for transversally elliptic operators on a compact foliated manifold.
In this paper we consider a singular elliptic equation involving the GJMS (Graham-Jenne-Mason-Sparling) operator of order k on n-dimensional compact Riemannian manifold with 2k<n. Mutiplicity and nonexistence results are established.
It was conjectured by Bott-Grove-Halperin that a compact simply connected Riemannian manifold $M$ with nonnegative sectional curvature is rationally elliptic. We confirm this conjecture under the stronger assumption that $M$ has entire Grauert tube,i.e.,$M$ is a real analytic Riemannian manifold that has a unique adapted complex structure defined on the whole tangent bundle $TM$.
In this short note, we consider gradient estimates for positive solutions to the following nonlinear elliptic equation on a complete Riemannian manifold: $$Δu+cu^α=0,$$ where $c, α$ are two real constants and $c\neq 0$.
In the present paper we prove Liouville-type theorems: non-existence theorems for some complete Riemannian almost product manifolds and special mappings of complete Riemannian manifolds which generalize similar results for compact manifolds.
In this paper, we study eigenvalue of linear fourth order elliptic operators in divergence form with Dirichlet boundary condition on a bounded domain in a compact Riemannian manifolds with boundary (possibly empty) and find a general inequality for them. As an application, by using this inequality, we study eigenvalues of this operator on compact domains of complete submanifolds in a Euclidean space.
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.
Let $(M^{n+1}, g)$ be a compact Riemannian manifold with smooth boundary B and nonnegative Bakry-Emery Ricci curvature. In this paper, we use the solvability of some elliptic equations to prove some estimates of the weighted mean curvature and some related rigidity theorems. As their applications, we obtain some lower bound estimate of the first nonzero eigenvalue of the drifting Laplacian acting on functions on B and some corresponding rigidity theorems.
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.
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。
A Lie group as a 4-dimensional pseudo-Riemannian manifold is considered. This manifold is equipped with an almost product structure and a Killing metric in two ways. In the first case Riemannian almost product manifold with nonintegrable structure is obtained, and in the second case - a pseudo-Riemannian one. Each belongs to a 4-parametric family of manifolds, which are characterized geometrically.
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).
A Sasaki-like almost contact complex Riemannian manifold is defined as an almost contact complex Riemannian manifold which complex cone is a holomorphic complex Riemannian manifold. Explicit compact and non-compact examples are given. A canonical construction producing a Sasaki-like almost contact complex Riemannian manifold from a holomorphic complex Riemannian manifold is presented and called an $S^1$-solvable extension.
A Riemannian almost product manifold with integrable almost product structure is called a Riemannian product manifold. In the present paper the natural connections on such manifolds are studied, i.e. the linear connections preserving the almost product structure and the Riemannian metric.
The authors define a SNS (semi-nearly-sub)-Riemannian connection on nearly sub-Riemannian manifolds and study the geometric properties of such a connection, and obtain the natures of horizontal curvature tensors between horizontal sub-Riemannian connection and SNS-Riemannian connection. The authors further investigate the geometric characteristics of the projective SNS-Riemannian connection, and obtain a necessary and sufficient condition for a nearly sub-Riemannian manifold being projectively flat.
We prove conformal versions of the local decomposition theorems of de Rham and Hiepko of a Riemannian manifold as a Riemannian or a warped product of Riemannian manifolds. Namely, we give necessary and sufficient conditions for a Riemannian manifold to be locally conformal to either a Riemannian or a warped product. We also obtain other related de Rham-type decomposition theorems. As an application, we study Riemannian manifolds that admit a Codazzi tensor with two distinct eigenvalues everywhere.
The authors first in this paper define a semi-symmetric metric non-holonomic connection (called in briefly a semi-sub-Riemannian connection) on sub-Riemannian manifolds, and study the relations between sub-Riemannian connections and semi-sub-Riemannian connections. An invariant under a connection transformation $\nabla\rightarrow D$ is obtained. The authors then further deduce a sufficient and necessary condition that a sub-Riemannian manifold associated with a semi-sub-Riemannian connection is flat, and derive that a sub-Riemannian manifold with vanishing curvature with respect to semi-sub-Ri…
In this note we address a notion of sublaplacians of sub-Riemannian manifolds. In particular for fat sub-Riemannian manifolds we answered the sublaplacian question proposed by R. Montgomery.
The purpose of this paper is to study anti-invariant Riemannian submersions from Kenmotsu manifolds onto Riemannian manifolds. Several fundamental results in this respect are proved. The integrability of the distributions and the geometry of foliations are investigated. We proved that there do not exist (anti-invariant) Riemannian submersions from Kenmotsu manifolds onto Riemannian manifolds such that characteristic vector field ξ is a vertical vector field. We gave a method to get horizontally conformal submersion examples from warped product manifolds onto Riemannian manifolds. Furthermore, …
B. Sahin [9] introduced the notion of anti-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds. In the present paper we extend the notion of anti-invariant and Lagrangian Riemannian submersions (a special anti-invariant Riemannian submersion) to the case of locally conformal Kaehler manifolds. We discuss the geometry of foliation and obtain some decomposition theorems for the total manifold of such submersions.
$(N(k),ξ)$-semi-Riemannian manifolds are defined. Examples and properties of $(N(k),ξ)$-semi-Riemannian manifolds are given. Some relations involving ${\cal T}_{a}$-curvature tensor in $(N(k),ξ)$-semi-Riemannian manifolds are proved. $ξ$-${\cal T}_{a}$-flat $(N(k),ξ)$-semi-Riemannian manifolds are defined. It is proved that if $M$ is an $n$-dimensional $ξ$-${\cal T}_{a}$-flat $(N(k),ξ)$-semi-Riemannian manifold, then it is $η$-Einstein under an algebraic condition. We prove that a semi-Riemannian manifold, which is $T$-recurrent or $T$-symmetric, is always $T$-semisymmetric, where $T$ is any t…
As a generalization of anti-invariant Riemannian submersions, we introduce anti-invariant Riemannian maps from almost Hermitian manifolds to Riemannian manifolds. We give examples and investigate the geometry of foliations which are arisen from the definition of an anti-Riemannian map. Then we give a decomposition theorem for the source manifold of such maps. We also find necessary and sufficient conditions for anti-invariant Riemannian maps to be totally geodesic and show that every pluriharmonic Lagrangian Riemannian map, which is a special anti-invariant Riemannian map, from a Kähler manifo…
We extract a new class of paracontact paracomplex Riemannian manifolds arising from certain cone construction, call it para-Sasaki-like Riemannian manifold and give explicit examples. We define a hyperbolic extension of a paraholomorphic paracomplex Riemannian manifold, which is a local product of two Riemannian spaces with equal dimensions, showing that it is a para-Sasaki-like Riemannian manifold. If the starting paraholomorphic paracomplex Riemannian manifold is complete Einstein with negative scalar curvature then its hyperbolic extension is a complete Einstein para-Sasaki-like Riemannian …
In this paper, we study slant submanifolds of Riemannian manifolds with Golden structure. A Riemannian manifold $(\tilde{M},\tilde{g},φ)$ is called a Golden Riemannian manifold if the $(1,1)$ tensor field $φ$ on $\tilde{M}$ is a golden structure, that is $φ^{2}=φ+I$ and the metric $\tilde{g}$ is $φ-$ compatible. First, we get some new results for submanifolds of a Riemannian manifold with Golden structure. Later we characterize slant submanifolds of a Riemannian manifold with Golden structure and provide some non-trivial examples of slant submanifolds of Golden Riemannian manifolds.
We prove that the L^2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold induces a metric space structure. As the L^2 metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove several results on the exponential mapping and distance function of a weak Riemannian metric on a Hilbert/Frechet manifold. The statements are analogous to, but weaker than, what is known in the case of a Riemannian metric on a finite-dimensional manifold or a strong Riemannian metric on a Hilbert manifold.
In this paper, we define and study CR-lightlike submanifolds of a golden semi-Riemannian manifold. We investigate some properties of geodesic CR-submanifolds of a golden semi-Riemannian manifold. Moreover, we obtain many interesting results for totally geodesic and totally umbilical CR-submanifolds on a golden Riemannian manifold. A non-trivial example of CR-lightlike submanifold of a golden semi-Riemannian manifold is also constructed.
The largest class of Riemannian almost product manifolds, which is closed with respect to the group of the conformal transformations of the Riemannian metric, is the class of the conformal Riemannian P-manifolds. This class is an analogue of the class of the conformal Kähler manifolds in almost Hermitian geometry. The main aim of this work is to obtain properties of manifolds of this class with connections, whose curvature tensors have similar properties as the Kähler tensors in Hermitian geometry.
Three-dimensional Riemannian manifolds associated with locally conformal Riemannian product manifolds
本文研究带四次幂单位(1,1)型张量结构的三维黎曼流形，证明其关联近乘积流形属于Naveira分类中的局部共形黎曼乘积流形，并给出结构平行条件、爱因斯坦类及曲率性质，以超曲面为例。
Definition of $({\cal T}_{a},{\cal T}_{b})$-pseudosymmetric semi-Riemannian manifold is given. $({\cal T}_{a},{\cal T}_{b})$-pseudosy mmetric $(N(k),ξ)$-semi-Riemannian manifolds are classified. Some results for ${\cal T}_{a}$-pseudosymmetric $(N(k),ξ)$-semi-Riemannian manifolds are obtained. $({\cal T}_{a},{\cal T}_{b},S^{\ell})$-pseudosymmetric semi-Riemannian manifolds are defined. $({\cal T}_{a},{\cal T}_{b},S^{\ell})$-pseudosymmetric $(N(k),ξ)$-semi-Riemannian manifolds are classified. Some results for $(R,{\cal T}_{a},S^{\ell})$-pseudosymmetric $(N(k),ξ)$-semi-Riemannian manifolds are ob…
We prove that Riemannian $Spin(7)$ holonomy manifolds carry octonionic-Kähler structure.
We describe, by their holonomy groups, all complete simply connected irreducible non-locally symmetric pseudo-Riemannian SpinC manifolds which admit parallel spinors. So we generalize the Riemannian SpinC case and the pseudo-Riemannian Spin one.
In this paper, we consider critical maps of a horizontal energy functional for maps from a sub-Riemannian manifold to a Riemannian manifold. These critical maps are referred to as subelliptic harmonic maps. In terms of the subelliptic harmonic map heat flow, we investigate the existence problem for subelliptic harmonic maps. Under the assumption that the target Riemannian manifold has non-positive sectional curvature, we prove some Eells-Sampson type existence results for this flow when the source manifold is either a step-2 sub-Riemannian manifold or a step-r sub-Riemannian manifold whose sub…
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