(1.1) Some of the main results described in [Report] are the following (in rough terms; notations and precise references will be given below): (1) A map (f>:{M,g)-+(N,h) between Riemannian manifolds which is continuous and of class L\\ is harmonic if and only if it is a critical point of the energy functional. (2) Let (M, g) and (N, h) be compact, and <̂ 0: (M, g) -> (N, h) a map. Then ^0 can be deformed to a harmonic map with minimum energy in its homotopy class in the following cases: (a) Riem ' f t ^0; (b) dim M = 2 and n2(N) = 0. (3) Any map 0O: S m -> S can be deformed to a harmonic map provided m ^ 7. More generally, suitably restricted harmonic polynomial maps can be joined to provide harmonic maps between spheres. (4) The homotopy class of maps of degree 1 from the 2-torus T to the 2-sphere S has no harmonic representative, whatever Riemannian metrics are put on T and S. (5) If in (2) M has a smooth boundary, then various Dirichlet problems have solutions in case (a) and (b); and also when the boundary data is sufficiently small.

Another Report on Harmonic Maps

Quaternions.- Linear algebra over the quaternions.- Projective spaces.- Vector bundles.- The mean curvature sphere.- Willmore Surfaces.- Metric and affine conformal geometry.- Twistor projections.- Backlund transforms of Willmore surfaces.- Willmore surfaces in S3.- Spherical Willmore surfaces in HP1.- Darboux transforms.- Appendix: The bundle L. Holomorphicity and the Ejiri theorem.

/pdf/conformal-geometry-of-surfaces-in-s4-and-quaternions-5cssbo32sy.pdf

Conformal Geometry of Surfaces in S4 and Quaternions

The paper develops the fundamentals of quaternionic holomorphic curve theory. The holomorphic functions in this theory are conformal maps from a Riemann surface into the 4-sphere, i.e., the quaternionic projective line. Basic results such as the Riemann-Roch Theorem for quaternionic holomorphic vector bundles, the Kodaira embedding and the Pluecker relations for linear systems are proven. Interpretations of these results in terms of the differential geometry of surfaces in 3- and 4-space are hinted at throughout the paper. Applications to estimates of the Willmore functional on constant mean curvature tori, respectively energy estimates of harmonic 2-tori, and to Dirac eigenvalue estimates on Riemannian spin bundles in dimension 2 are given.

/pdf/quaternionic-holomorphic-geometry-plucker-formula-dirac-2d87l5qvc5.pdf

Quaternionic holomorphic geometry: Plucker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori

/pdf/a-differential-equation-arising-from-scalar-curvature-7w480loy5q.pdf

A differential equation arising from scalar curvature function

Isospectral deformations II: Trace formulas, metrics, and potentials

Willmore Surfaces with a Duality in SN(1)

https://projecteuclid.org/download/pdf_1/euclid.jmsj/1240319338

Compact minimal submanifolds of a sphere with positive Ricci curvature

https://projecteuclid.org/download/pdf_1/euclid.jmsj/1239888172

A negative answer to a conjecture of conformal transformations of Riemannian manifolds

In [2], Bishop and O'Neill studied some properties of warped product. We shall review them. Let B and F be Riemannian manifolds and f a positive C oo, function on B. Consider the product manifold B • with projections 7z: B x F ~ B and r/: B x F ~ F. The warped product M = B x yF is the manifold B x F with the Riemannian structure such that HX[J2= 117r,x[le+(f(Tzm)) z ) / , x r J 2 for all tangent vector X~TmM. Identifying TraM with T,(m)B+ T,(m)F, we say that X is horizontal if X~T~(m)B and Y is vertical if Y~T,(,,)F. B is isometric to ~/l(p) for p~F with isometry 7r and t/-l(p) for peF is totally geodesic in M. So the gradient offoTr on M may be identified with the gradient o f f on B. We denote by G the gradient o f f and fo ~z. We denote by ( , ) (resp. ( , )) the Riemannian metric on F(resp. B or M) and by D(resp. V) the covariant differentiation of F(resp. B or M).

A Construction of Non-Flat, Compact Irreducible Riemannian Manifolds Which are Isospectral But Not Isometric.

We construct some compact hypersurfaces of constant scalar curvature in a sphere with cohomogeneity 1([2]).

Norio Ejiri

Papers

Willmore Surfaces with a Duality in SN(1)

Compact minimal submanifolds of a sphere with positive Ricci curvature

A negative answer to a conjecture of conformal transformations of Riemannian manifolds

A Construction of Non-Flat, Compact Irreducible Riemannian Manifolds Which are Isospectral But Not Isometric.

Some compact hypersurfaces of constant scalar curvature in a sphere