By J. Madore

This can be an advent to noncommutative geometry, with particular emphasis on these instances the place the constitution algebra, which defines the geometry, is an algebra of matrices over the complicated numbers. functions to user-friendly particle physics also are mentioned. This moment variation is punctiliously revised and contains new fabric on fact stipulations and linear connections plus examples from Jordanian deformations and quantum Euclidean areas. just some familiarity with usual differential geometry and the speculation of fiber bundles is believed, making this publication available to graduate scholars and newbies to this box.

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Nun{2" , 2"}. But since E(:Z:m) --+ E(:z:o) (m --+ 00) this implies that m. The contradiction proves the lemma. ) on M, and let :Z:1,:Z:2 be distinct strict relative minima of E. Then E possesses a third critical point :Z:3 distinct from :Z:1I :Z:2. 5") P = {p EC([O, 1]; M) I p(O) = :Z:1, p(1) = Z2}. 39 II. Unstable minimal surfaces Moreover and Z3 is unstable in the sense that Z3 is not a relative minimum of E. 11 13 > sup{E(zd, E(Z2)}. e. K{:I O. 9. = By definition of 13 = there is pEP such that sup E(z) xEp Applying the map P.

Convex We moreover assume that M is also closed in H and that the topologies induced on M by H and T coincide. 2") = E(:Z:0)+~d2E(:Z:o)(:Z:-:Z:0, :Z:-:Z:o)+o(I:Z:-:Z:ol1-), (dE(:z:) , :z: - y) = d 2E(:z:o)(:Z: - :Z:o, :z: - y) + 0(1:z: - :Z:011-) E(:z:) for all :z:, y E M such that 111 - :z:IH :::; I:z: - :Z:oIH· 53 II. 1: i) In case M =T =H our assumptions simply mean that E E C 2 (H). 2) implicitly we assume that at any critical point = 0, Vz EM. 2) would suffice. 2) holds for z rt. M. 2) at any critical point Zo E M there exists a bounded self-adjoint linear map a(zo): H -+ H such that Let Ho be the kernel of a(zo), and let subs paces of H such that H_, H+ be maximal a(zo)-invariant (a(zo)e,e) < 0 , VeE H_, eto, (a(zo)e,e) > 0 , VeE H+, eto.

1) in terms of 2d V kE~}. 3) Finally, for z E M let JJ 2,.. 2,.. 4) E(z) = D(X(z» = _1 167r °° , 2 h(z(¢» - 'Y(z~¢ »1 d¢ d¢', . SIn 2 (¢ -¢) -2- denote the Douglas - Dirichlet integral of X(z) = h('Y 0 z), cf. Nitsche [1;§§ 310 311]. 42 A. urfaces. 1: For any a E IR the set {Z E M· I E(z) ~ a} is compact with respect to the CO-topology. 2: Suppose {zm} is a sequence in M such that IZm(O)1 ~ c uniformly. Suppose E(zm) ~ c uniformly. Then a subsequence Zm -+ Z in LII (IR) and either oc z(¢I)== const (mod 211"), or Zm and Z -+ Z uniformly (m -+ 00), E M.