A FORMULA FOR THE MEAN CURVATURE OF AN IMPLICIT REGULAR SURFACE

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1 STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume XLVI, Number 3, September 001 A FORMULA FOR THE MEAN CURVATURE OF AN IMPLICIT REGULAR SURFACE Abstract. In this paper we will ind a ormula or the absolute value o the mean curvature o an implicit regular surace (S) (, y, ) a, epressed in terms o the partial derivatives o the unction. 1. Introduction The most used ormulas or the Gaussian curvature or or the mean curvature o a regular surace are those that are epressed locally in terms o the coeicients o the irst and second undamental orms. However or an implicit regular surace (S) (, y, ) a there eists a ormula or the Gaussian curvature epressed in terms o the partial derivatives o the unction, that is, y K 1 y yy y y 4 y y 0 In this paper we are going to prove a similar ormula or the absolute value o the mean curvature o an implicit regular surace. ormula where For the mean curvature H o a regular surace S we have the ollowing local H 1 (1) eg F + ge EG F () E r u r u r u, F r u r v, G r v r v r v are the coeicients o the irst undamental orm and e ( r u, r v, r uu ) EG F, ( r u, r v, r uv ), g ( r u, r v, r vv ) EG F EG F 77

2 are the coeicients o the second undamental orm with respect to the local parametriation r : U S, compatible with the orientation o the surace. Let V R 3 be an open set, : V R be a dierentiable unction and a Im be a regular value o. It is well known that S 1 (a) is an orientable regular surace. For p S, then one o the partial derivatives (p), y (p), (p) is non ero, at least. I (p) 0, or instance, then, according to the implicit unction theorem, the last variable can be unically epressed by means o the irst two variable and y. In other words the regular surace S 1 (a) is locally, around the point p, the graph o a unction (, y), (, y) U, where U is a conveniently chosen open set. Thereore the mapping r : U S, r(, y) (, y, (, y)) is a local parametriation o S at p, namely (, y, (, y)) a, (, y) U. This is the type o local parametriation that we are going to use or all over this paper. It is very easy to see that r r y 1 which means that the local parametriation r : U S, r(, y) (, y, (, y)) o S at p is compatible with the orientation o S i (p) > 0 and o course uncompatible i (p) < 0. In any case the relation holds. eg F + ge H EG F (3). The main ormula In this section we will prove the already anounced ormula or the absolute value o the mean curvature o an implicit regular surace. Theorem.1. Let V R 3 be an open set, : V R be a smooth unction and a Im be a regular value o the. For the absolute value o the mean curvature H o the implicit regular surace (S) (, y, ) a, at the point p S, we have the ollowing ormula H 1 [ ( (Hess ), )], (4) where is the gradient o, is the Laplace s operator and Hess is the Hessian o, all o them being considered at the point p. 78

3 A FORMULA FOR THE MEAN CURVATURE OF AN IMPLICIT REGULAR SURFACE Proo. Assuming that or p 1 (a) we have (p) 0, it ollows that S is locally, around the point p, the graph o a unction (, y), (, y) U and consider the above stated local parametriation r : U S, r(, y) (, y, (, y)). The coeicients o the two undamental orms are H E 1 +, F, G 1 + y y e 1+, y + y 1+, g + y yy 1+ + y eg F + ge EG F (1 + ( ) ) yy y y + (1 + y ). (5) [1 + + y] 3/ Because (, y, (, y)) a, (, y) U, it ollows that + 0 From relations (6) we get y y yy y + y 0 [ (,y,(,y)) [ (,y,(,y)) (,y,(,y)) [ (,y,(,y)) y(,y,(,y)) (,y,(,y)) that is Replacing the partial derivatives, y,, y, yy the ormula (5) we obtain y y. ] + ] 3 y y y+y ] 3 yy yy+ y. 3 H (6) (7) given by the relations (6), (7)in ( y + )( + ) 5 y ( y y y + y ) 5 + ( + )( yy y y + y ) 5 [ + y + ] 3/ 3 [ y y + y y + y y y y y y + yy y y + y + 4 yy y 3 y + ] y y 3 5 ( y + + y y + yy + yy y y + ) y 3 ( yy + ) + y ( + ) + ( + yy ) (Hess )(, ) + + y yy + 3 where (Hesss )(, y ) (, y, ) y yy y y y 79

4 yy y y y y y Thereore or the absolute value o the mean curvature wee have H 1 ( + yy + ) + y ( + yy + + ( + yy + ) (Hess )(, ) 3 1 ( + + )( + + ) (Hess )(, ) y yy 3 1 (Hess )(, ) 3 1 [ ( )]. (Hess ), Corollary.. I V R 3 is an open set, : V R is a smooth harmonic mapping and a Im is a regular value o, then or the absolute value o the mean curvature o the implicit regular surace (S) (, y, ) a we have the ollowing ormula: 1 H (Hess )(, ). (8) 3 3. Eample It is well know that the locus o the orthogonal projections o the center o the ellipsoid (E) a + y b + c 1 on its tangent planes is the so called pedal surace o E, that is the regular surace S {(, y, ) R 3 ( + y + ) a + b y + c }\{0}. We will compute the absolute value o the mean curvature o the pedal surace o E in its points. For this purpose consider p ( 0, y 0, 0 ) S, the unction : R 3 \{0} R, (, y, ) ( + y + ) a b y c and observe that S 1 (0). 80

5 A FORMULA FOR THE MEAN CURVATURE OF AN IMPLICIT REGULAR SURFACE The partial derivatives o irst and second order o are y 4( + y + ) a 4y( + y + ) b y 4( + y + ) c 4( + y + ) + 8 a y y 8y 8 yy 4( + y + ) + 8y b y y 8y 4( + y + ) + 8 c. Thereore in the points (, y, ) o the regular surace S we have 4(a 4 + b 4 y +c 4 ), or equivalent (a 4 +b 4 y +c 4 ) 1/. Observe that 0 in all the points o the surace S 1 (0). Thereore the critical set o doesn t intersects the level set S 1 (0), this being o course an argument on the regularity o S. On the other hand 0( + y + ) (a + b + c ) and (Hess)(, ) yy y y y y y (4( + y + ) + 8 a )[16 ( + y + ) 16a ( + y + ) + 4a 4 ]+ +(4( + y + ) + 8y b )[16y ( + y + ) 16b y ( + y + ) + 4b 4 y ]+ +(4( + y + ) + 8 c )[16 ( + y + ) 16c ( + y + ) + 4c 4 ]+ +16y[4( + y + ) a ][4y( + y + ) a y]+ +16[4( + y + ) a ][4( + y + ) c ]+ +16y[4y( + y + ) a y][4( + y + ) c ] 48( + y + )(a 4 + b 4 y + c 4 ). Replacing all o these values considered in p, in the ormula (4), we obtain H S1 (p) 1 4(a b4 y 0 + c4 0( 0 )1/ 0 + y ) (a + b + c ) 48( 0 + y )(a4 0 + b4 y 0 + c4 0 ) 4(a b4 y 0 + c4 0 ) 4( 0 + y ) (a + b + c ) (a b4 y 0 + c4 a 0 + b y 0 + c 0 0 )1/ a b4 y 0 + c4 1 a + b + c. 0 a b4 y 0 + c4 0 81

6 Reerences [Ca] Carmo, M., do, Dierentiable Geometry o Curves and Suraces, Prentice-Hall, New Jersey, [Fe] Fédenko, A., Recueil D eercises de Géométrie Diérentielle Éditions MIR Moscou, 198. [Gr] Gray, A., Modern Dierential Geometry o Curves and Suraces, CRC Press, [Pi] Pintea, C., Geometrie. Elemente de Geometrie Analitică. Elemente de Geometrie Dierenţială a Curbelor şi Supraeţelor, Presa Universitară Clujeană, 001. Babeş-Bolyai University, Faculty o Mathematics, Str. M. Kogălniceanu 1, 3400 Cluj-Napoca, Romania address: cpintea@math.ubbcluj.ro 8