Principal curvature estimates for the convex level sets of semilinear elliptic equations
Sun-Yung Alice Chang Xi-Nan Ma Paul Yang
We give a positive lower bound for the principal curvature of the strict convex level sets of harmonic functions in terms of the principal curvature of the domain boundary and the norm of the boundary gradient. We also extend this result to a class of semi-linear elliptic partial differential equations under certain structure condition.
keywords: semilinear elliptic equation. Curvature estimate level sets
Positivity of Paneitz operators
Xingwang Xu Paul C. Yang
In this paper, we give two results concerning the positivity property of the Paneitz operator-- a fourth order conformally covariant elliptic operator. We prove that the Paneitz operator is positive for a compact Riemannian manifold without boundary of dimension at least six if it has positve scalar curvature as well as nonnegative $Q-$curvature. We also show that the positivity of the Paneitz operator is preserved in dimensions greater than four in taking a connected sum.
keywords: connected sum. Paneitz opertaor Eigenvalues

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