Algebraic Curves: An Introduction to Algebraic Geometry by William Fulton

By William Fulton

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Thus M\ll = ยข. P 46 Let us then define 111 by 111 : ~p E M~t Sp ~ 0}. First, we assume that 111 is not empty. Then 111 is a non empty open part of M and in each point p of Ill, we know that (Vh)p = 0. The classical Pick-Berwald theorem then implies that 111 is an open part of a nondegenerate ellipsoid or hyperboloid. Thus detS is a constant different from zero on 111" The continuity of detS then implies that fit = M. Finally, we may assume that S = 0 on the whole of M. Thus by Proposition 2, we can suppose that M is given by the equation z -- P(x,y), where P is a polynomial of degree at most k + 1, and that the canonical affine normal vector field is given by (0,0,1).

Amer. Math. Soc. 210, 75-106 (1975). 2. , Rigoli, M. and Woodward, minimal immersions of S 2 into CP n. 3. M. : On conformal Math. Ann. 279, 599-620 L. M. (1988). : Minimal immersions of S 2 and RP 2 into CP n with few higher order singularities. To appear in Math. Proc. Camb. Phil. Soc. 4. Bolton, J. and Woodward, forms. M. : On immersions of surfaces into space Soochow J. of Mathematics 5. Bolton, J. M. immersions with St-symmetry. 6. Bolton, J. M. 14, 11-31 (1988). : On the Simon conjecture for minimal Math.

Then it follows from Lemma 5 that a = 0 and ,11 = ,l 2. Thus, we have (i). Therefore, we may assume that detS = 0. But then we know that there exists an eigenvector u of S with eigenvalue zero. Again there are two possibilities. a. h(u,u) -- 0. In this case, we can find a vector v, such that h(v,v) = 0 and h(u,v) = 1. Using the equation of Ricci, we then obtain that h(Sv,u) = h(v,Su) = 0. Hence Sv has no component in the direction of v. Thus, we have (iii). b. h(u,u) \$ 0. Here, we may assume, by taking - ~ as normal, that h(u,u) = 1.