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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.
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
In this paper, we derive a priori interior Hessian estimates for Lagrangian mean curvature equation if the Lagrangian phase is supercritical and has bounded second derivatives.
We have an idea on the influence of a nonlinear term (tending to 0) on the prescribed scalar curvature equation to have an uniform estimate.
We prove an existence result for the prescribed Ricci curvature equation for certain doubly warped product metrics on $mathbb{S}^{d_1+1}times mathbb{S}^{d_2}$, where $d_i geq 2$. If $T$ is a metric satisfying certain curvature assumptions, we show th
In this note, we study Liouville type theorem for conformal Gaussian curvature equation (also called the mean field equation) $$ -Delta u=K(x)e^u, in R^2 $$ where $K(x)$ is a smooth function on $R^2$. When $K(x)=K(x_1)$ is a sign-changing smooth func