We introduce two flow approaches to the Loewner--Nirenberg problem on comapct Riemannian manifolds $(M^n,g)$ with boundary and establish the convergence of the corresponding Cauchy--Dirichlet problems to the solution of the Loewner--Nirenberg problem. In particular, when the initial data $u_0\in C^{4,α}(M)$ is a solution or a strict subsolution to the equation $(1.1)$, the convergence holds for both the direct flow $(1.3)-(1.5)$ and the Yamabe flow $(1.10)$. Moreover, when the background metric satisfies $R_g\geq0$, the convergence holds for any positive initial data $u_0\in C^{2,α}(M)$ for the direct flow; while for the case the first eigenvalue $λ_1<0$ for the Dirichlet problem of the conformal Laplacian $L_g$, the convergence holds for $u_0>v_0$ where $v_0$ is the largest solution to the homogeneous Dirichlet boundary value problem of $(1.1)$ and $v_0>0$ in $M^{\circ}$ (the interior of $M$). We also give an equivalent description between the existence of a metric of positive scalar curvature in the conformal class of $(M,g)$ and $\displaystyle\inf_{u\in C^1(M),\,u\not\equiv 0\,\text{on}\,\partial M}Q(u)>-\infty$, where $Q$ is the energy functional (see $(1.8)$) of the second type Escobar-Yamabe problem.
We introduce a flow approach to the generalized Loewner-Nirenberg problem $(1.5)-(1.7)$ of the $σ_k$-Ricci equation on a compact manifold $(M^n,g)$ with boundary. We prove that for initial data $u_0\in C^{4,α}(M)$ which is a subsolution to the $σ_k$-Ricci equation $(1.5)$, the Cauchy-Dirichlet problem $(3.1)-(3.3)$ has a unique solution $u$ which converges in $C^4_{loc}(M^{\circ})$ to the solution $u_{\infty}$ of the problem $(1.5)-(1.7)$, as $t\to\infty$.
We show for $k \geq 2$ that the locally Lipschitz viscosity solution to the $σ_k$-Loewner-Nirenberg problem on a given annulus $\{a < |x| < b\}$ is $C^{1,\frac{1}{k}}_{\rm loc}$ in each of $\{a < |x| \leq \sqrt{ab}\}$ and $\{\sqrt{ab} \leq |x| < b\}$ and has a jump in radial derivative across $|x| = \sqrt{ab}$. Furthermore, the solution is not $C^{1,γ}_{\rm loc}$ for any $γ> \frac{1}{k}$. Optimal regularity for solutions to the $σ_k$-Yamabe problem on annuli with finite constant boundary values is also established.
We consider the problem of finding on a given Euclidean domain $Ω$ of dimension $n \geq 3$ a complete conformally flat metric whose Schouten curvature $A$ satisfies some equation of the form $f(λ(-A)) = 1$. This generalizes a problem considered by Loewner and Nirenberg for the scalar curvature. We prove the existence and uniqueness of such metric when the boundary $\partialΩ$ is a smooth bounded hypersurface (of codimension one). When $\partialΩ$ contains a compact smooth submanifold $Σ$ of higher codimension with $\partialΩ\setminusΣ$ being compact, we also give a `sharp' condition for the di…
We develop a new approach to the conformal geometry of embedded hypersurfaces by treating them as conformal infinities of conformally compact manifolds. This involves the Loewner--Nirenberg-type problem of finding on the interior a metric that is both conformally compact and of constant scalar curvature. Our first result is an asymptotic solution to all orders. This involves log terms. We show that the coefficient of the first of these is a new hypersurface conformal invariant which generalises to higher dimensions the important Willmore invariant of embedded surfaces. We call this the obstruc…
In this paper, we prove the concavity of the Renyi entropy power for nonlinear diffusion equation (NLDE) associated with the Laplacian and the Witten Laplacian on compact Riemannian manifolds with non-negative Ricci curvature or $CD(0,m)$-condition and on compact manifolds equipped with time dependent metrics and potentials. Our results can be regarded as natural extensions of a result due to Savaré and Toscani \cite{ST} on the concavity of the Renyi entropy for NLDE on Euclidean spaces. Moreover, we prove that the rigidity models for the Renyi entropy power are the Einstein or quasi-Einstein …
In this paper we derive both local and global geometric inequalities on general Riemannnian and Finsler manifolds and prove generalized Caffarelli-Kohn-Nirenberg type and Hardy type inequalities on Finsler manifolds, illuminating curvatures of both Riemannian and Finsler manifolds influence geometric inequalities.
An optimal Loewner-type systolic inequality and harmonic one-forms of constant norm
给出了闭黎曼流形X上一个新的最优收缩不等式，它推广了包括C.洛纳我们用X的Abel-Jacobi映射A_X的几何刻划了边界相等的情形。对于一个极值度量，映射A_X是一个具有极小纤维的黎曼浸入到一个平坦环面上。我们用A.- M. Bergé和J. Martinet。给定一个闭流形X，它允许一个浸入F到它的Jacobi环面T^{b_1（X）}，我们构造X上所有实现不等式中的相等的度量。虽然可以在F的纤维上选择任意固定体积的度量，但水平空间是使用J. Moser构造体积保持流的多参数版本来选择的。
We study the existence of conformal metrics on non-compact Riemannian manifolds with non-compact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature on the boundary. These metrics are constructed on smooth manifolds obtained by removing d-dimensional submanifolds from certain n-dimensional compact spaces locally modelled on generalized solid cones. We prove the existence of such metrics if and only if d>(n-2)/2. Our main theorem is inspired by the classical results by Aviles-McOwen and Loewner-Nirenberg …
Nirenberg and Spencer posed the question whether the germ of a compact complex submanifold in a complex manifold is determined by its infinitesimal neighborhood of finite order when the normal bundle is sufficiently positive. To study the problem for a larger class of submanifolds, including free rational curves, we reformulate the question in the setting of families of submanifolds and their infinitesimal neighborhoods. When the submanifolds have no nonzero vector fields, we prove that it is sufficient to consider only first-order neighborhoods to have an affirmative answer to the reformulate…
Given a finite collection of $C^1$ complex vector fields on a $C^2$ manifold $M$ such that they and their complex conjugates span the complexified tangent space at every point, the classical Newlander-Nirenberg theorem gives conditions on the vector fields so that there is a complex structure on $M$ with respect to which the vector fields are $T^{0,1}$. In this paper, we give intrinsic, diffeomorphic invariant, necessary and sufficient conditions on the vector fields so that they have a desired level of regularity with respect to this complex structure (i.e., smooth, real analytic, or have Zyg…
Let $(M; g)$ be a smooth compact Riemiannian manifold without boundary and $g_{k}$ be a metric conformal to $g$. Suppose $vol(M; g_{k})+||R_{k}||_{L^{p}(M;g_{k})} < C$, where $R_{k}$ is the scalar curvature and $p > \frac{n}{2}$. We will use the 3-circle theorem and the John-Nirenberg inequality to study the bubble tree convergence of $g_{k}$.
We investigate when the Chevalley-Eilenberg differential of a complex Lie algebroid on a manifold with boundary admits a Hodge decomposition. We introduce the concepts of Cauchy-Riemann structures, elliptic and non-elliptic boundary points and Levi-forms, which we use to define the notion of q-convexity. We show that the Chevalley-Eilenberg complex of an elliptic, q-convex Lie algebroid admits a Hodge decomposition in degree q. This generalizes the well-known Hodge decompositions for the exterior derivative on real manifolds and the delbar-operator on q-convex complex manifolds. We establish t…
研究了具有负截面曲率的流形中极值域的几何和拓扑。极值区域是超定椭圆问题（简称OEP）存在正解的区域。我们考虑两种类型的OEP。首先，我们研究了Hadamard流形中这类区域的窄性质，并刻画了无穷远处的边界。我们给出了一个上界的豪斯多夫维数的边界在无穷远和如何域行为在无穷远。这显示了与奇异Yamabe问题的有趣关系。随后，我们将重点放在双曲空间$\mathbb H ^n$的极值域。将显示对称性和有界性属性。在一定意义上，我们将Levitt-Rosenberg定理\cite{LR}推广到OEP，这表明它与$\mathbb H ^n$中的常中曲率超曲面有很强的联系。特别是，我们能够证明Berestycki-Caffarelli-Nirenberg猜想在一定的假设下，无论是在边界上无穷远的极值域或OEP本身。并给出了双曲空间极值区域上解的高度估计。
We study the regularity of the solutions of second order boundary value problems on manifolds with boundary and bounded geometry. We first show that the regularity property of a given boundary value problem $(P, C)$ is equivalent to the uniform regularity of the natural family $(P_x, C_x)$ of associated boundary value problems in local coordinates. We verify that this property is satisfied for the Dirichlet boundary conditions and strongly elliptic operators via a compactness argument. We then introduce a uniform Shapiro-Lopatinski regularity condition, which is a modification of the classical…
