Some Result On Complete F-minimal Hypersurfaces In R~n×S~1(a)

In this paper we consider f-minimal hypersurfaces in the product space Rn × S1(a)where (Rn, e-fdμ) is the standard Gaussian space. The main content is divided into two parts. In the first part, by introducing a global smooth function a (the angle function of the hypersurface and S1(a)), we derived a few interesting differential identities of Simons type by which we can prove some rigidity results; In the second part, we study the stability properties of some standard examples. The main theorems obtained are as follows:Theorem 0.1 (See Theorem 3.2-3.6 in Chapter Three) Let x: Mn → Rn x S1 (a)be a complete oriented f-minimal hypersurface properly immersed in (Rn × S1(a),g).(1) If the angle function a is constant, then either α = 0 and x(Mn) = Σn-1×S1 (a)with Σn-1 being a complete self-shrinker in Rn, or α=1 and x(Mn) = Rn × {s0}, a slice.(2) If a does not change its sign and |h| ∈ Lf2, then either x(Mn) = Σn-1 × S1(a), or x(Mn)=Rn×{s0}.(3) If |▽h| ∈ Lf2 and there is a constant c, 0 < c < 1, such that 2|▽α|2 ≤ c|▽h|2 and|h|2 ≤ 1 + α2, then either h≡ 0 and x(Mn) = Rn × {s0} or Rn-1×S1(a), or |h|2≡1 and x(Mn) = Sk((?)) ×Rn-k-1 × S1(a), 1 ≤ k ≤ n - 1.(4) If |h|2 < 2α2 and there is a constant c, 0 < c < 1, such that |▽α|2 ≤ c|▽h|2, then either x(Mn) = Rn × {s0} or Rn-1 × S1(a).(5) If |h|2 ≤ 3α2 - 1 and either |h|2 = const or |▽h| ∈ Lf2, then x(Mn) = Rn × {s0}.(6) If 1/2(1 - α2 - c) ≤ |h|2 ≤≤1/2(1 - α2 + c) with c = (?), then either x(Mn) = Rn × {s0} or Rn-1×S1(a) or x(Mn) = Sk((?)) ×Rn-k-1×S1(a) for 1 ≤k≤n -1.Theorem 0.2 (See Theorem 4.1 in Chapter Four) The slice inclusion i: Rn × {s0}→Rn × S1(a) is the only stable f-minimal hypersurface in (Rn × S1(a),g).This dissertation is divided into four chapters:Chapter One is an introduction presenting the background and the main results.Chapter Two is the preliminary material, which is divided into two sections: In Section One we introduce the definitions and notations which we will use later; While in Section Two we give some typical examples of f-minimal hypersurfaces in Rn×S1(a).In Chapter Three we give some Somons type identities about the f-minimal hyper-surfaces in Rn × S1(a) and then prove some rigidity results.In Chapter Four we discuss the stability properties of typical examples, then get the theorem 4.1.