Critical Point Theory and Submanifold Geometry by Richard S. Palais

By Richard S. Palais

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Then the new coordinate system (x, y), defined by dx = a(p) dp, dy = b(q) dq, gives the fundamental forms as in the theorem. 3. , u = 2~, is a solution for the Sine-Gordon equation. 13) I I = 2 sin u ds dr. 14) (s, t) are called the Tchebyshef coordinates. The Sine-Gordon equation becomes ust : sin u. 6. Hilbert T h e o r e m . There is no isometric immersion o f the simply connected hyperbolic 2-space H 2 into R 3. PROOF. Suppose H z can be isometrically immersed in R 3. Because A1A2 --- - 1 , there is no umbilic points on H 2, and the principal directions gives a global orthonormal tangent frame field for H 2.

17) reduces to an ordinary differential equation, u" = a s i n h u , which always has periodic solution. But it was proved by Hsiang and Lawson in [HL] that there are only countably many immersed minimal tori in S 3, that admit an Sl-action. 18) may not close up to a solution on M (the period problem is more complicated than for the torus case). Let (M, ds 2) be a closed surface with constant curvature k, and dg 2 = e2~'ds2. Suppose ( M , dg 2) is isometrically immersed in N3(c) with constant mean curvature H , and Q is the associated holomorphic quadratic differential.

So e l ( x ) , . . , e n ( x ) are tangent to M for x C M . L e t a ) l , . . , w A ( e e ) = 5AB. 6), we have CAWs +coAeB = O, and 2. Local Geometry of Submanifolds 35 13 Set 02 A ~ £A~2 A Since en+l = X , we have den+l = E 02n+ i 1 @ ei = dX = E i o2 i @ e i . i i So con+ 1 = coi. By the Gauss equation we have a i = --(W? q-1 n COn+ J 1 -- -- --a2~ +1 A cojn-t-1 = --w i A w j = --wi A w j. So M has constant sectional curvature - 1 . From now on we will let H n denote M with the induced metric from R n'l .

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