球面上預設均曲率之超曲面

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(2) Hypersurfaces with Prescribed Mean Curvature in Spheres .

(3) . ! "#$%&'()*+,)- E-mail: yjhsu@math.nctu.edu.tw! ! Keywords: Mean curvature, hypersurface, spheres n+1

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(5) 1. Introduction !"#$%&'()* Let M and N be complete Riemannian +,-. /0123456 789:;<= > ?@A>(BCD8E)*+, F manifolds- dim(M)= n, dim(N) = n+1. Let H be a preassigned smooth function defined on !" N. We consider the problem of prescribed mean curvature, that is, find conditions on H GH H so that there exists an embedding Y from M into N whose mean curvature is the given Abstract function H (see [Y]). For N being the Let H be a given smooth real valued function Euclidean space and M being a sphere, defined on the (n+1)-sphere. The problem of significant works in this direction have been prescribed mean curvature is to find certain studied in [BK] and [TW]. Based on the conditions on H so that there exists a theory of elliptic partial differential equation, hypersurface M of spherical type whose Treibergs and Wei showed the existence and mean curvature is the given function H.The uniqueness of the problem of prescribed purpose of this report is to show the mean curvature if H decays faster than the existence of solutions of a related evolution mean curvature of two concentric spheres equation. Using the Schauder fixed point [TW]. For both of N and M being spheres, theorem, we consider a linearized equation, using the theory of elliptic partial differential and prove that under the assumption that equation [GT], we also showed the existence the given function H satisfies certain of the problem of prescribed mean curvature conditions, a apriori estimates of maximum if H satisfies certain growth conditions. norm and gradient of the solutions for this In this report, we want to show the linearized equation hold. It follows that the existence of the solution for a evolution flow solution of this evolution equation exists.. 1.

(6) equation We show that if H satisfies the following three conditions :. related to the problem of prescribed mean curvature. Let Y be a map from the n-sphere into the (n+1)-sphere which is given by. 1. Y(X) =. u. X+. (H 1) H (. E n+ 2 ,. 1+ α for α < −c 1. 1+ u2 1 + u2 where X ∈ S n , E n+ 2 = ( 0,0,L ,0,1).. (H 2) H (. Yt = ( H − N ) N , where N is the unit normal and N !is the mean curvature of Y.. for α > c 2 (H 3) Hα (. 1 + u2. 1 aij u ij + b in ℜ + × S n , n 1 + u + ∇u. u. = u 0 on S n ,. t=0. α 1+ α. 2. E n+ 2 ) < α ,. 1 1+ α. 2. X+. 1. X +. α 1 +α. 2. E n+ 2 ) > α ,. α. E n+ 2 ) 1 +α2 1+ α 2 α 1 α >− H( X+ En+ 2 ), 2 1+ α 1+ α 2 1 +α2. Then the corresponding quasilinear parabolic equation is. linearized equation hold. Theose conditions are essentially growth condition. We then can state the main result of this report as follows. 2. 2. 2. X +. for some positive constants c1 and c2, then a priori estimates of the solutions for this. We consider the following related evolution equation. ut =. 1. (*) 2. where a ij = (1 + u 2 + ∇u )δ ij - u i u j , 1 1+ u 2 u ∇u 2 2 n 1 + u + ∇u. 2. b = u (1 + u ) − 1+ u. 2. Theorem. Let H be a function defined on the (n+1)-sphere satisfying (H1), (H2) and (H3). Then there exists a solution u in C2,1 of the evolution equation (*).. 2. 2. 2. 1 + u + ∇u H .. To show the existence of solutions for such a equation, we need make a apriori estimates for the solutions of the following linearized. 2. Proof of Theorem We separate the proof into two parts. In the first part we show that the maximum estimate follows from (H1) and (H2). In the second part we show that the gradient estimate follows from (H3).. 1 + u2. 1 ut = aij u ij + b in ℜ + × S n , 2 2 1 + u + ∇u n u. t=0. n. = u 0 on S , 2. 2. where a ij = (1 + u + ∇u )δ ij - u i u j , 1 1 +u 2 b =u+ n 1 + u 2 + ∇u 2. We only show that the lower bound estimate follows from (H1), similar arugument show that upper bound estimate follows from (H2). Let x be the infimum of u. We may assume that x< 0. Claim that: x is not. 2. σ ( − 1 + u 2 1 + u 2 + ∇u H + 2. (1 + u )u − +. 2 1 1 + u2 u ∇ u 2 n 1 + u 2 + ∇u. 1 1 +u 2 u ), n 1 + u 2 + ∇u 2. σ ∈ [0,1].. 2.

(7) less than -c1. Suppose that x is less than -c1 , by using (H1), we have. 0 ≥ ut = ≥ .−. 3. Final Comments In this report we show that the evolution equation related to the problem of prescribed mean curvature has a solution. A natural question which should be discussed in the next work is that whether this solution converge to a solution of prescribed mean curvature. That is, the behavior of the stated state must be studied. Since we have the elliptic type result, it is possible that the stated state is just our desired hypersphere. What would be particularly interesting here is an answer to the following question: Can one find any singularities at finite time or infinite time when one of these conditions (H1), (H2) and (H3) false ? What is the behavior of the singular set ? Is there any results analogue to well known results, such as Ricci flow, mean curvature flow etc. ? these problem are also interesting to us.. 1+ u2. 1 aij uij + b 1 + u 2 + ∇u n 2. 1 1 u + σ (− (1 + u 2 )H + (1 + u 2 )u + u ) n n. >0 we get a contraction. To make the gradient estimate, after rescale the time variable t and omit the lower order term, which does not play any rule in the following argument, we may assume that. 4. References. u t = aij u ij + b in ℜ + × S n , u. t=0. = u 0 on S n ,. [BK] I.. Bakelman and B.Kantor, Estimates for solutions of quasilinear elliptic equations connected with problems of geometry in the large, Mat. Sb. 91(133), 1972, Math. USSR-Sb. 20(173), 348-363. [GT] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of the second order, Springer, New York, 1977.. where a ij = (1 + u )δ ij - u i u j , 2. 2. b = σ ( − 1 + u 2 1 + u 2 + ∇ u H ),. σ ∈ [0, n]. Let. ϕ=. ∇u. 2. . (1 + u 2 ) n Then the following inequality follows from the evolution equation of this function 3. ϕ t ≤ aijϕ ij − O( ∇ u )(H u +. [TW] A. E. Treibergs and S. W. Wei, Embedded hyperspheres with prescribed mean curvature, J. Differential Geometry, 18, 1983, 513-521. [Y] S. T. Yau, Seminar on differential Geometry, Annals of Math. Studies, No. 102, problem section, Princeton University Press, 1982, 669-706.. u H) 1+ u2. 2. + O(∇ϕ , ∇ u ), where the second term in the right hand side is negative. The gradient estimate then follows from the maximum principle .. . 3.

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