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Register a new userPrescribing integral curvature equation
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Meijun Zhu
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In this paper we formulate new curvature functions on $mathbb{S}^n$ via integral operators. For certain even orders, these curvature functions are equivalent to the classic curvature functions defined via differential operators, but not for all even orders. Existence result for antipodally symmetric prescribed curvature functions on $mathbb{S}^n$ is obtained. As a corollary, the existence of a conformal metric for an antipodally symmetric prescribed $Q-$curvature functions on $mathbb{S}^3$ is proved. Curvature function on general compact manifold as well as the conformal covariance property for the corresponding integral operator are also addressed, and a general Yamabe type problem is proposed.
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For $pin (1,2]$ and a bounded, convex, nonempty, open set $Omegasubsetmathbb R^2$ let $mu_p(bar{Omega},cdot)$ be the $p$-capacitary curvature measure (generated by the closure $bar{Omega}$ of $Omega$) on the unit circle $mathbb S^1$. This paper shows that such a problem of prescribing $mu_p$ on a planar convex domain: Given a finite, nonnegative, Borel measure $mu$ on $mathbb S^1$, find a bounded, convex, nonempty, open set $Omegasubsetmathbb R^2$ such that $dmu_p(bar{Omega},cdot)=dmu(cdot)$ is solvable if and only if $mu$ has centroid at the origin and its support $mathrm{supp}(mu)$ does not comprise any pair of antipodal points. And, the solution is unique up to translation. Moreover, if $dmu_p(bar{Omega},cdot)=psi(cdot),dell(cdot)$ with $psiin C^{k,alpha}$ and $dell$ being the standard arc-length element on $mathbb S^1$, then $partialOmega$ is of $C^{k+2,alpha}$.
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