A ''sup+ c inf'' inequality for the equation -delta u = [ by Bartolucci D.

By Bartolucci D.

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Denote the components of P with 1 J respect to X 1 , ••• , Xm. Then 47 eO~)gij -u~*h) .. s1J.. )o .. 1 1J q ( 1 - 2 oiq) 2 Now - e(0) ~ IJ. 1 IJ q therefore s1J.. 2) o .. •. ,l'n=l = 0. m Example 3. 3. 2 A: M - 1f 0 is a harmonic morphism, as in Section 1. •• , v = 0. n n+ m Example 3. 3. 3 In Proposition 2. 2. •• p, 1 i t- j, where n = "Ye. e. the direct sum of S. with E J field to the i soparametric family. 48 ~, where t is the unit normal vector 4 Equivariant theory 4. 1 Maps which are equivariant with respect to isoparametric functions.

In Example 2. 1. 4 there is one focal variety which is isometric to R m-p. 1. 5 there are two focal varieties, one is isometric to sP-1 and the other is isometric to sQ-1. 6 there is one focal variety isometric to HP-1. 1) where u, v E R, X, Y, z E F, F is one of R, 0:, quaternions or Cayley numbers, and is conjugation on F. Then f = F 18 3 11 + 1 is an isoparametric function of 35 degree 3 (all such arise in this way [6)). The focal varieties are antipodal, and are isometric to the projective plane P 2 (F), embedded as the standard Veronese minimal submanifold in the sphere.

Then g is a generalized isoparametric function. For is a hyperc surface of constant mean curvature in N, then n- 1 (N ) has constant mean curvature c in M [ 15). Example 2. 4. 7 Let N is a harmonic Riemannian submersion, and N Consider the tangent bundle TM of a Riemannian manifold ( M,g). n be the projection map TI: TM - M. Define the metric G on TM to be the Sasaki metric [ 35 ) , and write D for the Levi-Ci vita connection on TM • The metric G and connection D have the properties that (i) the horizontal lift of any geodesic ofM is a geodesic of TM.

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