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Local gradient estimates of p -harmonic functions, 1 / H -flow, and an entropy formula

Brett Kotschwar, Lei Ni (2009)

Annales scientifiques de l'École Normale Supérieure

In the first part of this paper, we prove local interior and boundary gradient estimates for p -harmonic functions on general Riemannian manifolds. With these estimates, following the strategy in recent work of R. Moser, we prove an existence theorem for weak solutions to the level set formulation of the 1 / H (inverse mean curvature) flow for hypersurfaces in ambient manifolds satisfying a sharp volume growth assumption. In the second part of this paper, we consider two parabolic analogues of the p -harmonic...

Local reflexion spaces

Jan Gregorovič (2012)

Archivum Mathematicum

A reflexion space is generalization of a symmetric space introduced by O. Loos in [4]. We generalize locally symmetric spaces to local reflexion spaces in the similar way. We investigate, when local reflexion spaces are equivalently given by a locally flat Cartan connection of certain type.

Local well-posedness of solutions to 2D magnetic Prandtl model in the Prandtl-Hartmann regime

Yuming Qin, Xiuqing Wang, Junchen Liu (2025)

Applications of Mathematics

We consider the 2D magnetic Prandtl equation in the Prandtl-Hartmann regime in a periodic domain and prove the local existence and uniqueness of solutions by energy methods in a polynomial weighted Sobolev space. On the one hand, we have noted that the x -derivative of the pressure P plays a key role in all known results on the existence and uniqueness of solutions to the Prandtl-Hartmann regime equations, in which the case of favorable P ...

Localization of basic characteristic classes

Dirk Töben (2014)

Annales de l’institut Fourier

We introduce basic characteristic classes and numbers as new invariants for Riemannian foliations. If the ambient Riemannian manifold M is complete, simply connected (or more generally if the foliation is a transversely orientable Killing foliation) and if the space of leaf closures is compact, then the basic characteristic numbers are determined by the infinitesimal dynamical behavior of the foliation at the union of its closed leaves. In fact, they can be computed with an Atiyah-Bott-Berline-Vergne-type...

Locally conformally Kähler metrics on Hopf surfaces

Paul Gauduchon, Liviu Ornea (1998)

Annales de l'institut Fourier

A primary Hopf surface is a compact complex surface with universal cover ℂ 2 - { ( 0 , 0 ) } and cyclic fundamental group generated by the transformation ( u , v ) ↦ ( α u + λ v m , β v ) , m ∈ ℤ , and α , β , λ ∈ ℂ such that ∣ α ∣ ≥ ∣ β ∣ > 1 and ( α - β m ) λ = 0 . Being diffeomorphic with S 3 × S 1 Hopf surfaces cannot admit any Kähler metric. However, it was known that for λ = 0 and ∣ α ∣ = ∣ β ∣ they admit a locally conformally Kähler metric with parallel Lee form. We here provide the construction of a locally conformally Kähler metric with parallel Lee form for all primary Hopf surfaces of class 1 ( λ = 0 ). We also show...

Locally symmetric immersions

José Carmelo González-Dávila, Lieven Vanhecke (1999)

Czechoslovak Mathematical Journal

We use reflections with respect to submanifolds and related geometric results to develop, inspired by the work of Ferus and other authors, in a unified way a local theory of extrinsic symmetric immersions and submanifolds in a general analytic Riemannian manifold and in locally symmetric spaces. In particular we treat the case of real and complex space forms and study additional relations with holomorphic and symplectic reflections when the ambient space is almost Hermitian. The global case is also...

Logarithmic Surfaces and Hyperbolicity

Gerd Dethloff, Steven S.-Y. Lu (2007)

Annales de l’institut Fourier

In 1981 J. Noguchi proved that in a logarithmic algebraic manifold, having logarithmic irregularity strictly bigger than its dimension, any entire curve is algebraically degenerate.In the present paper we are interested in the case of manifolds having logarithmic irregularity equal to its dimension. We restrict our attention to Brody curves, for which we resolve the problem completely in dimension 2: in a logarithmic surface with logarithmic irregularity 2 and logarithmic Kodaira dimension 2 , any...

Lorentzian geometry in the large

John Beem (1997)

Banach Center Publications

Lorentzian geometry in the large has certain similarities and certain fundamental differences from Riemannian geometry in the large. The Morse index theory for timelike geodesics is quite similar to the corresponding theory for Riemannian manifolds. However, results on completeness for Lorentzian manifolds are quite different from the corresponding results for positive definite manifolds. A generalization of global hyperbolicity known as pseudoconvexity is described. It has important implications...

Lorentzian similarity manifolds

Yoshinobu Kamishima (2012)

Open Mathematics

An (m+2)-dimensional Lorentzian similarity manifold M is an affine flat manifold locally modeled on (G,ℝm+2), where G = ℝm+2 ⋊ (O(m+1, 1)×ℝ+). M is also a conformally flat Lorentzian manifold because G is isomorphic to the stabilizer of the Lorentzian group PO(m+2, 2) of the Lorentz model S m+1,1. We discuss the properties of compact Lorentzian similarity manifolds using developing maps and holonomy representations.

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