By Leo Sario, Mitsuru Nakai, Cecilia Wang, Lung Ock Chung (auth.)

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**Additional resources for Classification Theory of Riemannian Manifolds: Harmonic, quasiharmonic and biharmonic functions**

**Example text**

The (radial) boundary of neighborhood does. The corresponding transformation is sense-reversing and conformal, and the function interior of p But at such points~ about every din with Thus the function hm(t ) = h(t) - h(tm(t)) is harmonic on all of W. In the same manner as for chosen~ hm(t ) ~ 0 Furthermore, on W, 8h/~8 = 0 and on the m = i, we conclude that, with h(t) Rv mn on Smk k = k(~) is symmetric about and at the points on dmk properly dmk. that do not lie on any slit. 11. without comon points letting W s OHp.

3. p Proof. subregion with If R e 0~, < if is unique up to an additive constant. and L1 are denoted by y. For PO and Pl" We claim: is valid for every g(x,y) m = mLu820" g(x,y) on 2 - 20' 2. N on R and a regular and a regular subregion hence m(l - ~) ! g on ~ R - 20 . A ~N R e 0~. and Conversely, L0 = O~ m(l - ~2) < g w ~ O p take a Green's function containing ~0 c 2, to an__~d 0 NG. 0~ The equality 20 fortiori~ The function corresponding Equality of THEOREM. R [i]). , de Rham [i]). f~*d~ > 0, for otherwise 8~/~n ~ 0 liberty of integrating along ~ on Since ~, and ~ > 0 ~ ~ 0.

Courant-Hilbert [11). ONG < N :> OI_]l for all The inclusion h + c c HP. hence also to N N OpiP c O ~ h ~ HB(W0) , N-space R0, we obtain ~ N = 2, the D, h = const, and conclude that h e HB giving rise to an consider the Riemann surface obtained by puncturing the above The function h(z) = -log izl W belongs to and W0 R~ HP(W0) , On the other hand, the origin is a removable singularity hence for every G N > 2, h ~ HP, ,--7' +r h(O) is immediate, every 2-manifold HP(R0). to an For N-space N > 2.