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**Extra info for A Mathematical Introduction to String Theory: Variational Problems, Geometric and Probabilistic Methods**

**Example text**

Thus, if z =

4) h for Re(s) > 0. e(A) > 0 we have the equality T{s)\-3 = H dtta-xe-tX. 5) is just the Mellin transform of the function e~tX. Thus for Re(s) > f I"'dtt-Hr'e-" T(S)CA(S)= dt)t3-Hr'e-tA, = ( / dt + / Jo J\ where tr'e~tA = £''e~ tX «. 3) the integral over [1, oo) is bounded by a constant times the integral J^° dt t3~i e~'A°, Ao being the smallest eigenvalue different from zero, so it defines an entire function of 6. , the an(x) being certain A-dependent scalar functions. 7) the integral over [0,1] splits into the sum of the following two integrals: h{s) := Vols(M) / dt r> t'^Oit1^) V / ^ .

M'. g € Ml,l > 3, can be uniformized by a smooth hyperbolic metric g-i on S, by Poincare's uniformization theorem and its extension to the case of continuous metrics; see [J2] for details. 2. u*(g) is then also of class H1, and therefore differs from g-i by a positive function of class Hl. Conversely, multiplying any g' 6 Ml_x by a positive Hl function yields an element g £ Ml, for / > 3. 5: Ml is homeomorphic to Tp x DQ +1 X C', and M is diffeomorphic to Tp x Vo x C. ^1, and Tp are contractible for / > 3.