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If a reai n-dimensional compact Riemannian manifold (M, g) satisfies the Sobolev inequality (J\4>\^)~ \ + \eC\M), then the heat kernel H satisfies the estimate 0 < H < C(t~nl2 constant C which depends only on n and S. + 1) with a positive 3. Existence of admissible Hermitian metrics Let £ be a rank r reflexive sheaf on an n-dimensional compact Kahler manifold (X, u). We take a finite cover { Ua } and local resolutions • • • —► £ ^ a —► E^a —► £ v | r / —> 0 of the dual sheaf £v by holomorphic vector bundles, which admits the following commutative diagram.

We consider the heat equation for Hermitian metrics of the holomorphic vector bundle E on {Xk,Uk,e) taking the above constructed h as the initial metric. ^ / i - 1 = - ( v ^ T A f c , £ F - X(S)I). k,e means the trace taken with respect to the metric wjt>e. Since every thing is now in the smooth category, the result of Donaldson [4] says that (2)fc has a solution untill infinite time. The curvature tensor satisfies the following. 2)fc Jxk ~ Jy£Xh ~ Jy€Xk Jy€Xk where Qt>£ and Hk,e(t,x,y) and its heat kernel.

1 dl>dY, 3E" . / d l > \ . [W >,(4)^ - ^(r- jdj + ^ 5 ( ^ )fi,P^-Kw{*-«a*| ~~arH~arr/l tf*(D*-|*(f)* UH]-- [T",[/"] = A tit. ds dE" . (dY,"\ ~dT8*\-arj dE" . / d E " \ l . 1) yield \T", U") + [TH,U»] + [ r " , UH] + [W,W] H HH = ,U,W) ) = T(UHT(U ,W)H,TH)--T(T - T(T H + T(UH,TH)--T(T",U + T(U",W) T(T",W). ) Furthermore, if V e V we have (VVU»,T»)&. =Gai,(a,)\n [ * , ( § £ ) +r? 7 (a 8 ) ( § £ ) ] « ? = G a (a,)r? ) ( ^ ) V7 = G „ ( * . ) ( — ) V" = «e(n u"K> and (VFtf», I*)*. &yU",T") a. = G^(^)F7 ^ ( ^ ) ] (A -WJ*M%)U = 0.

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3-D Shapes by Marina Cohen


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