Generalized external cone condition for domains in Riemannian manifolds

Separable p-harmonic functions in a cone and related quasilinear equations on manifolds

In considering a class of quasilinear elliptic equations on a Riemannian manifold with nonnegative Ricci curvature, we give a new proof of Tolksdorf's result on the construction of separable $p$-harmonic functions in a cone.

A singular Yamabe problem on manifolds with solid cones

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 …

锥体和扭曲积流形上的热点

Hot spots on cones and warped product manifolds

研究了扭曲积流形上热方程热点的极值问题及长期行为，证明了Rauch猜想在特定条件下成立，并揭示了谱间隙对长期行为的影响。

Sweeping process by prox-regular sets in Riemannian Hilbert manifolds

In this paper, we deal with sweeping processes on (possibly infinite-dimensional) Riemannian Hilbert manifolds. We extend the useful notions (proximal normal cone, prox-regularity) already defined in the setting of a Hilbert space to the framework of such manifolds. Especially we introduce the concept of local prox-regularity of a closed subset in accordance with the geometrical features of the ambient manifold and we check that this regularity implies a property of hypomonotonicity for the proximal normal cone. Moreover we show that the metric projection onto a locally prox-regular set is sin…

Riemannian geometry of Kahler-Einstein currents

We study Riemannian geometry of canonical Kahler-Einstein currents on projective Calabi-Yau varieties and canonical models of general type with crepant singularities. We prove that the metric completion of the regular part of such a canonical current is a compact metric length space homeomorphic to the original projective variety, with well-defined tangent cones. We also prove a special degeneration for Kahler-Einstein manifolds of general type as an approach to establish the compactification of the moduli space of Kahler-Einstein manifolds of general type. A number of applications are given f…

Local Geometry of Singular Real Analytic Surfaces

Let V be a compact real analytic surface with isolated singularities embedded in $R^N$, and assume its smooth part is equipped with a Riemannian metric that is induced from some analytic Riemannian metric on $R^N$. We prove: 1. Each point of V has a neighborhood which is quasi-isometric (naturally and 'almost isometrically') to a union of metric cones and horns, glued at their tips. 2. A full asymptotic expansion, for any $p\in V$, of the length of $V\cap\{q:\dist(q,p)=r\}$ as r tends to zero. 3. A Gauss-Bonnet Theorem, saying that horns do not contribute an extra term, while cones contribute …

Scattering resonances on truncated cones

We consider the problem of finding the resonances of the Laplacian on truncated Riemannian cones. In a similar fashion to Cheeger--Taylor, we construct the resolvent and scattering matrix for the Laplacian on cones and truncated cones. Following Stefanov, we show that the resonances on the truncated cone are distributed asymptotically as Ar^n + o(r^n), where A is an explicit coefficient. We also conclude that the Laplacian on a non-truncated cone has no resonances away from zero.

Degree of mobility for metrics of lorentzian signature and parallel (0,2)-tensor fields on cone manifolds

Degree of mobility of a (pseudo-Riemannian) metric is the dimension of the space of metrics geodesically equivalent to it. We describe all possible values of the degree of mobility on a simply connected n-dimensional manifold of lorentz signature. As an application we calculate all possible differences between the dimension of the projective and the isometry groups. One of the main new technical results in the proof is the description of all parallel symmetric (0,2)-tensor fields on cone manifolds of signature $(n-1,2).

$\sigma_k$-Loewner-Nirenberg问题在黎曼流形上对于$k<\frac{n}{2}$

The $σ_k$-Loewner-Nirenberg problem on Riemannian manifolds for $k<\frac{n}{2}$

本文证明在紧致黎曼流形上，当锥满足代数条件（如$k<n/2$的Gårding锥）时，完全非线性Loewner-Nirenberg问题存在Lipschitz粘性解，并同时获得Dirichlet问题存在唯一性及共形度量的存在性。

Helffer-Nourrigat 锥的分层

Stratification of the Helffer-Nourrigat Cone

本文提出通过分层方法对Helffer-Nourrigat锥进行去奇异化，揭示其在亚黎曼几何中的相空间作用，并展示相关C*-代数的可解性。

Nash-type equilibria on Riemannian manifolds: a variational approach

Motivated by Nash equilibrium problems on 'curved' strategy sets, the concept of Nash-Stampacchia equilibrium points is introduced via variational inequalities on Riemannian manifolds. Characterizations, existence, and stability of Nash-Stampacchia equilibria are studied when the strategy sets are compact/noncompact geodesic convex subsets of Hadamard manifoleds, exploiting two well-known geometrical features of these spaces both involving the metric projection map. These properties actually characterize the {non-positivity} of the sectional curvature of complete and simply connected Riemannia…

On the existence of solutions of the second boundary value problem for $p$-Laplacian on Riemannian manifolds

We obtain necessary and sufficient existence conditions for solutions of the boundary value problem $$ Δ_p u = f \quad \mbox{on } M, \quad \left. \left| \nabla u \right|^{p - 2} \frac{\partial u}{\partial ν} \right|_{ \partial M } = h, $$ where $p > 1$ is a real number, $M$ is a connected oriented complete Riemannian manifold with boundary, and $ν$ is the external normal vector to $\partial M$.

Energy scaling law for the regular cone

We consider a thin elastic sheet in the shape of a disk whose reference metric is that of a singular cone. I.e., the reference metric is flat away from the center and has a defect there. We define a geometrically fully nonlinear free elastic energy, and investigate the scaling behavior of this energy as the thickness $h$ tends to 0. We work with two simplifying assumptions: Firstly, we think of the deformed sheet as an immersed 2-dimensional Riemannian manifold in Euclidean 3-space and assume that the exponential map at the origin (the center of the sheet) supplies a coordinate chart for the w…

具有外部数据的分数阶各向异性Calderón问题

Fractional anisotropic Calderón problem with external data

本文在欧几里得空间中解决了二维及以上的分数阶各向异性Calderón问题，证明了部分外部Dirichlet--to--Neumann映射可确定黎曼度量至微分同胚，并给出了两种证明方法。

Injectivity Radius of Lorentzian Manifolds

Motivated by the application to spacetimes of general relativity we investigate the geometry and regularity of Lorentzian manifolds under certain curvature and volume bounds. We establish several injectivity radius estimates at a point or on the past null cone of a point. Our estimates are entirely local and geometric, and are formulated via a reference Riemannian metric that we canonically associate with a given observer $(p,T)$ --where $p$ is a point of the manifold and $T$ is a future-oriented time-like unit vector prescribed at $p$. The proofs are based on a generalization of arguments fro…