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Register a new userDifferential geometry in the theory of Hessian operators
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The paper introduces a new differential-geometric system which originates from the theory of $m$-Hessian operators. The core of this system is a new notion of invariant differentiation on multidimensional surfaces. This novelty gives rise to the following absolute geometric invariants: invariant derivatives of the surface position vector, an invariant connection on a surface via subsurface, curvature matrices of a hypersurface and its normal sections, $p$-curvatures and $m$-convexity of a hypersurface, etc. Our system also produces a new interpretation of the classic geometric invariants and offers new tools to solve geometric problems. In order to expose an application of renovated geometry we deduce an a priori $C^1$-estimate for solutions to the Dirichlet problem for $m$-Hessian equations.
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We give an explicit description of the full asymptotic expansion of the Schwartz kernel of the complex powers of $m$-Laplace type operators $L$ on compact Riemannian manifolds in terms of Riesz distributions. The constant term in this asymptotic expansion turns turns out to be given by the local zeta function of $L$. In particular, the constant term in the asymptotic expansion of the Greens function $L^{-1}$ is often called the mass of $L$, which (in case that $L$ is the Yamabe operator) is an important invariant, namely a positive multiple of the ADM mass of a certain asymptotically flat manifold constructed out of the given data. We show that for general conformally invariant $m$-Laplace operators $L$ (including the GJMS operators), this mass is a conformal invariant in the case that the dimension of $M$ is odd and that $ker L = 0$, and we give a precise description of the failure of the conformal invariance in the case that these conditions are not satisfied.
In this paper, we consider Hessian equations with its structure as a combination of elementary symmetric functions on closed Kahler manifolds. We provide a sufficient and necessary condition for the solvability of these equations, which generalize the results of Hessian equations and Hessian quotient equations.
In this note we obtain local and global Hessian estimates for the conjugate heat equation coupled with the Ricci flow. As an application, we extend Perelmans monotonicity formula of the $W$-entropy to the noncompact situation.
A new proof for stability estimates for the complex Monge-Amp`ere and Hessian equations is given, which does not require pluripotential theory. A major advantage is that the resulting stability estimates are then uniform under general degenerations of the background metric in the case of the Monge-Amp`ere equation, and under degenerations to a big class in the case of Hessian equations.
This paper has been withdrawn by the authors due to the fact that the webs considered in the paper are ``Veronese-like webs which are different from Veronese webs.
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