Some Properties Of Compact α-Hessian Manifold

In this paper, we define anα- Hessian manifold M. Then we introduce some invariants and calculate the first variation of the volume functional. By estimating the third derivative, we show a Bernstein property.The main theorems we get are:(1)Let M be a compact extremalα- Hessian manifold withα≠0. When the module of variant vector field is small enough and (n＋2)(n-1)/n~2＋n≤α≤n＋2/n,then M is maximal, i.e.∫M dV_t≤∫M dV.(2)Let M be a compact extremalα- Hessian manifold, there exists a positive constant K(n) depending only on the dimension n such that if∣α∣＞K(n), M must be E_n/Γ, whereΓis a subgroup of isometrics of E~n which acts freely and properly discontinuously on E~n.