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On a fourth order equation in 3-D

Xingwang Xu, Paul C. Yang (2002)

ESAIM: Control, Optimisation and Calculus of Variations

In this article we study the positivity of the 4-th order Paneitz operator for closed 3-manifolds. We prove that the connected sum of two such 3-manifold retains the same positivity property. We also solve the analogue of the Yamabe equation for such a manifold.

On a Fourth Order Equation in 3-D

Xingwang Xu, Paul C. Yang (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this article we study the positivity of the 4-th order Paneitz operator for closed 3-manifolds. We prove that the connected sum of two such 3-manifold retains the same positivity property. We also solve the analogue of the Yamabe equation for such a manifold.

On metrics of positive Ricci curvature conformal to M × 𝐑 m

Juan Miguel Ruiz (2009)

Archivum Mathematicum

Let ( M n , g ) be a closed Riemannian manifold and g E the Euclidean metric. We show that for m > 1 , M n × 𝐑 m , ( g + g E ) is not conformal to a positive Einstein manifold. Moreover, M n × 𝐑 m , ( g + g E ) is not conformal to a Riemannian manifold of positive Ricci curvature, through a radial, integrable, smooth function, ϕ : 𝐑 𝐦 𝐑 + , for m > 1 . These results are motivated by some recent questions on Yamabe constants.

On Schrödinger maps from T 1 to  S 2

Robert L. Jerrard, Didier Smets (2012)

Annales scientifiques de l'École Normale Supérieure

We prove an estimate for the difference of two solutions of the Schrödinger map equation for maps from T 1 to  S 2 . This estimate yields some continuity properties of the flow map for the topology of  L 2 ( T 1 , S 2 ) , provided one takes its quotient by the continuous group action of  T 1 given by translations. We also prove that without taking this quotient, for any t > 0 the flow map at time t is discontinuous as a map from 𝒞 ( T 1 , S 2 ) , equipped with the weak topology of  H 1 / 2 , to the space of distributions ( 𝒞 ( T 1 , 3 ) ) * . The argument relies in an essential...

On special Riemannian 3 -manifolds with distinct constant Ricci eigenvalues

Oldřich Kowalski, Zdeněk Vlášek (1999)

Mathematica Bohemica

The first author and F. Prufer gave an explicit classification of all Riemannian 3-manifolds with distinct constant Ricci eigenvalues and satisfying additional geometrical conditions. The aim of the present paper is to get the same classification under weaker geometrical conditions.

On the asymptotic geometry of gravitational instantons

Vincent Minerbe (2010)

Annales scientifiques de l'École Normale Supérieure

We investigate the geometry at infinity of the so-called “gravitational instantons”, i.e. asymptotically flat hyperkähler four-manifolds, in relation with their volume growth. In particular, we prove that gravitational instantons with cubic volume growth are ALF, namely asymptotic to a circle fibration over a Euclidean three-space, with fibers of asymptotically constant length.

On the Curvature and Heat Flow on Hamiltonian Systems

Shin-ichi Ohta (2014)

Analysis and Geometry in Metric Spaces

We develop the differential geometric and geometric analytic studies of Hamiltonian systems. Key ingredients are the curvature operator, the weighted Laplacian, and the associated Riccati equation.We prove appropriate generalizations of the Bochner-Weitzenböck formula and Laplacian comparison theorem, and study the heat flow.

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