By J.N. Coldstream

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Randol, private communication). 47 5. Disks in Constant Curvature Space Forms PROOF:Here we have C,(t) = c o s h 6 t S,(t) = ( s i n h G t ) / f i , On the one hand, the Cauchy-Schwarz inequality implies that for any C’ function d ( t )on [O, S] we have -1: 44’s:On the other hand, 1 5 ””{jjPs:-1y2. {j)2s:- $(a) = 0 implies that -j;44s:- = J;(42,2Xs:- = (n - 1) I)r, jo(42/2)c,s:-2 d Therefore (49) -(n - 1)2(~/4) I d 4’Sz-l I Jo#rzS:-l. It remains to note that (49) will imply (46). Indeed, given f then -(n - 1 ) ’ ( ~ / 4 ) 1 f 2 = -(n - E CF(B(G)), 1)2(~/4) Wdt and (46) follows by Rayleigh’s theorem (see Pinsky [l]).

Remark I : We note the contrast in the hypotheses of the two domain monotonicity theorems for eigenvalues, the addition of vanishing Neumann data requiring a complete partition of M. To employ this result in a fixed geometric setting, with good choices of Q,, . ,R , one requires a priori some knowledge of the decomposability of M . 3. DEFINITION 6. Let f: M -+ R E Co. Then the nodal set o f f is the set f-'[O], and a nodal domain off is a component on M\f-'[O]. COURANT'S NODAL DOMAIN THEOREM. Let (79) be our list of eigenvalues and {& b2,.

29 2. Ton One checks that the same eigenvalues are obtained, as when only admitting real-valued functions, with the same multiplicity. To obtain a collection of eigenfunctions on T we proceed as follows: Associate to the lattice I', the dual lattice, r*,given by r * =( y ~ R " : ( ~ , y ) ~ P f o r a l l x ~ r ) . Then r*is indeed a lattice of rank n, and to the above basis {ul, . , un} of r is associated the dual basis { w , , . , w,,} of r*,determined by = 8;. (wj,uk) where 8 j k is the Kronecker delta.