Tazawa, Yoshihiko

English

Hokkaido University Collection of Scholarly and Academic Papers

A REMARK ON A STAR-SHAPED HYPERSURFACE WITH CONSTANT REDUCED MEAN CURVATURE By Y oshihiko T AZA W A The purpose of the present paper is to give another proof of the following proposition proved by A. Aeppli: Proposition. A star-shaped hypersuぐface F (ωith re，学ect to a fixed point 0) in μn (n + 1)-dimensional Euclidean space Rn+l which has constant reduced mean curvature rHl (with res，ρect ω 0) is α hyper，ゅんere around 0 , where r=r(ρ) is the distance between 0 and PE F, and Hl = Hl (ρ) is the first mean curvature of F. Concerning the problem whether a closecl hypersurface with rHl = const becomes a hypersphere around 0 or not, Aeppli gave the a伍rmatlve answer in the cases ( 1) F is star-shaped (cf [1]1) Theorem 4 and Footnote 11日，( 2) rHl = 1 (cf [2] Proposition 1), ( 3) F is simple (i. e., without self-intersections) and 0 lies in the interior of F (cf [2] Proposition 1"). The author showed already that under a certain condition (which he called “ radial convexity") F cannot possess constant reduced mean curvature (cf [5]). In general, the problem of characterizing the hypersphere around 0 by constant reduced curvatures is not solved. The method of the proof used in this paper is based on “ reflection and sliding" due to A. D. Alexandrov (cf [3] and [4]). The author wishes to express here his sincere thanks to Professor Y 0-shie Katsuracla for her kind guidance. 1. Notations and Lemmas. Let F be a connected simple closed hypersurface of class C2 in an (n + 1)-dimensional Euclidean space Rn+ l, n 孟 2，and 0 be a 五xecl point in Rncl. Let F be star-shapecl with respect to 0 , i. e., there exists a bijective clifferentiable central projection from F to the unit hypersphere around 0 without critical points. Therefore 0 lies in the 1) Numbers in brackets refer to the references at the end of the paper .4 Remark on a Star-Shαped Hypersurface with Constant Reduced 1'/[，ωnC加でatuγe 123 interior of F, and n(ρ). r (ρ)<0， where n(ρ) is the inner normal of F at p and r(ρ) is the radius vector from 0 to p, and any ray issued from 0 meets F only once (cf [1] Footnote 10) and [4] S2). In [2], Aeppli uses the word “ star-shaped" in a slightly weaker sense. Let Hν =Hν(ρ) (ρε F) be the ぃth mean curvature H戸(nCJ-l .z= k1 ••• k" where k,/s are the principal curvatures. The トthe reduced mean curvature with respect to 0 is rνH，. 1n [2] , Aeppli showed the following two lemmas: Lemma 1. (1n a local expression of F: Z = xn I 1 = xn i 1 (x\ … , Xn)) (α) rH1 =c (c=const) is an e切りwhere absolutely elliptic (partial) 晶表rentialequation. (b) 1f ム>0 for all i= 1,… , n, then r'H,=c is 仰 absolutelyelliptic d江戸èrential equation. Lemma 2. Let F1 and F2 be tωo (regular) suぐfaces in contact at p both of ωhich are solutions of the absolutely elliptic (μrtial) dゆrentialequation (ザ、 second order) φ= 0 (FJ, F2 of class C2 ; φ=φ(x， Z, Zi' zu) of class CO in all variables x¥ x 2, ..., Znn and of class C1 in z , ZJ，・..， Zn ,J_ Then the intersection of F1 and F2 consists of a set N (ρεN) such that tlze ωntact at p between F1 and F2 is not semiアropeァ， or else F1 and 凡 coincide in a neighbourhood of p. Let Rp and E, be two mappings from R勾 i 1 onto itself de五ned as follows : ( 1) Rp is the re自ection with respect to a hyperplane P which passes through 0 , ( 2 )τ(E， (q)) is equal to the radius vector t r(q) for al qERη+ 1， that is, E, is the t-times homothetical extention with respect to 0 , where t isa posi圃tive number. 1t is clear that rνf王 is invariant under Rp and E, for al P and t , if we take always the inner normals. 2. Another proof of the proposition. Let P be a hyperplane which passes through O. For su伍ciently large positive number t , (凡oRp)(F) n F=。and F lies in the interior of the hypersurface (E,oRp ) (F). Let t decrease until (E,cRp ) (F) touches F for the 五rst time (consider a continuous function p(p) on the compact set F, such that ρ(ρ) = r(q)/r(p), where q is the unique point on (E,oRp ) (F) which lies on the ray issued from 0 through p, then the point at which ρtakes the minimum is a f?st common point such that (E,oRp ) (F) touches F). As we always consider inner normals, and because of the star-shapedness of (E,oRl') (F) and F, the contact of F and (凡oRp)(F) is positive and proper. Therefore by Lemma 1 (a) and Lemma 2, F and (E,oRp ) (F) coincide in a neighbourhood of a point of contact, i. e., the set of the points of contact is open. By continuity it is also closed. Therefore 124 Y. lazawa F=(EtoRp) (F)2l. Let q be a point in Fn P. Since Rp(q) = q, q and (EtoRp) (q)lie on the same ray issued from O. The star-shapedness of F yields q=(EtoRp) (q)because of F=(EtoRp) (F) , i. e. , q=Et(q) and t must coincide with 1, and Et = Et = identity. Therefore F= Rp(F), in other words, F is symmetric with respect to P. Since we can take P in any direction, F becomes a hypersphere around O. Q.E.D. The proposition holds good for the ぃth reduced mean curvature r"H川if 丸>0 for al i= 1, ・・・ ， n, by virtue of Lemma 1 (b) : Let F be the same as in Proposition (excφt rHt= c). If ム>0 for all i=l , "', n ωzd rν瓦 =c for a fixed l.i, then F is a hyper~ρherで around 0. Remark 1. In the proof above we stated that a closed (star-shaped) hypersurface F which is symmetric with respect to any hyperplane through o isa hypersphere around O. For n = 2 the proof of the anologuos asserｭtion is seen in [3]. For higher dimensions we can verify it without di伍cultyby induction (it is su伍cient if we show that the intersection of F and the hyperplane through 0 is an (n-1)-sphere around 0 with constant radius). Remark 2. The more generalization of the proposition is given by A. D. Alexandrov in [4] ~ 2 (Theorem 2). There he showed 五rst that c= 1 and concludεd that F is a hypersphere. References [1] A. AEPPLI: Einige Ahnlichkeits-und Symmetries舩ze f? diHerenzierbare Fl臘hen im Raum, Comment. Math. Helv目 33 (1959), 174-195. [2] A. AEPPLI: On the uniqueness of compact solutions for certain eliptic diferential equations, Proc. Amer. Math. Soc. 11 (1960), 826-832. [3] H. HOPF: Lectures on differential geometry in the large (Notes by ]. W. Gray), Stanford Univ. (1956) [4] A. D. ALEXANDROV: Uniqueness theorems for surfaces in the large 1, Amer. Math. Soc. Trans. Ser. 2, Vol. 21 (1962), 341-354. [ 5] Y. T AZA W A: A remark on a closed orientable hypersurface with constant reduced mean curvature, ]. Fac目 Sci. Hokkaido Univ. Ser. 1, Vol. 20, No. 3 (1968), 101-L08. Department of Mathematics, Hokkaido University. (Received March 14, 1970) 2) The coincidence of F and (Eto[{p) (F) is also obtained by the method based on the integral formula used in [1] 

A REMARK ON A STAR-SHAPED HYPERSURFACE WITH CONSTANT REDUCED MEAN CURVATURE

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A REMARK ON A STAR-SHAPED HYPERSURFACE WITH CONSTANT REDUCED MEAN CURVATURE

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