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On a fourth order equation in 3-D

Xingwang XuPaul 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 XuPaul 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.

Minimal surfaces in pseudohermitian geometry

Jih-Hsin ChengJenn-Fang HwangAndrea MalchiodiPaul Yang — 2005

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the p-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a characteristic curve, and as a quantity in terms of calibration geometry. As a differential equation, the p-minimal surface equation...

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