User fible - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T15:42:40Z http://mathoverflow.net/feeds/user/15474 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/61606/surfaces-of-constant-centro-affine-curvature/66814#66814 Answer by fible for surfaces of constant centro-affine curvature fible 2011-06-03T11:14:35Z 2011-06-03T11:34:55Z <p>You can see the book by Li-Simon-Zhao:"Global Affine differetial geometry of hypersurfaces. Berlin: Walter de Gruyter, 1993." I think yours definition of the constant centro-affine curvature surfaces means the ellptic affine spheres. Let M be a surface with constant centro-affine curvature 1, locally given by the graph $(x, f(x))$, then by the Legendre transformation relative to $f$, the equation is (with center at the orgin) $$\det(D^2 u)=(u)^{-4}.$$ The answer can be found in this book, due to Blaschke, Deicke, Calabi.</p> http://mathoverflow.net/questions/66748/how-do-control-the-boundary-regularity-of-the-legendre-transformation-domain-fro How do control the boundary regularity of the Legendre transformation domain from a convex function fible 2011-06-02T15:52:51Z 2011-06-02T15:58:59Z <p>Let f(x) be a strongly convex smooth function (its Hessian matrix is positive definite) defined in a convex domain D, introduce the Legendre transformation $$x=(x_1,...,x_n)\rightarrow (\xi_1,...,\xi_n),\xi_i=\frac{\partial f}{\partial x_i},$$ $$u(\xi_1,...,\xi_n)=x_i\xi_i-f$$ The Legendre transformation domain W is defined by: $$W=((\xi_1,...,\xi_n)|\xi_i=\frac{\partial f}{\partial x_i}, x\in D )$$ I want to know the regularity of the boundary of W, (can assume the domain W is bounded) what conditions to make the boundary $\partial W$ smooth or $C^2$?</p> http://mathoverflow.net/questions/15509/solutions-to-a-monge-ampere-equation-on-the-simplex/66466#66466 Answer by fible for Solutions to a Monge-Ampère equation on the simplex fible 2011-05-30T15:43:52Z 2011-05-30T15:43:52Z <p>Its useful to see the paper" [1] S.Y. Cheng, S.T. Yau. On the regularity of the solutions of the Monge-Amp$\grave{e}$re equation $\det(\frac{\partial^2 u}{\partial x_i\partial x_j})=F(x,u)$. Comm Pure Appl. Math, 1977, 30: 41-68."</p> <p>For special $\mu$, the hyperbolic affine sphere is a solution in a simplex with zero boundary value.</p> http://mathoverflow.net/questions/66748/how-do-control-the-boundary-regularity-of-the-legendre-transformation-domain-fro Comment by fible fible 2011-06-05T02:28:13Z 2011-06-05T02:28:13Z In my question, f(x) may be defined on the whole $\mathbb{R}^n$ (this situation is my interest), for example, the following function (known as hyperbolic affine hypersphere): $$f(x_1,...,x_n)=\frac{1}{x_1\cdots x_n}, x_i&gt;0, 1\leq i\leq n.$$ Choose suitable coordinates, this graph can be represented by another function $\tilde{f}$ defined on the whole $\mathbb{R}^n$, and the Legendre transformation domain of $\tilde{f}$ is a simplex.