[Paper Review] The regularity of harmonic maps into spheres and applications to Bernstein problems

Jürgen Jost, Yuanlong Xin+1 more|ArXiv.org|Dec 2, 2009

Geometric Analysis and Curvature Flows25 references3 citations

TL;DR

This paper establishes the regularity and a-priori estimates for weakly harmonic maps from Riemannian manifolds into spheres under the condition that their image avoids a neighborhood of a half-equator. By constructing compactly supported strictly convex functions on the complement of a half-equator, the authors derive new Bernstein-type theorems, proving that complete minimal hypersurfaces in Euclidean space or compact minimal hypersurfaces in spheres are affine planes or equatorial spheres if their Gauss maps omit such a neighborhood.

ABSTRACT

We show the regularity of, and derive a-priori estimates for (weakly) harmonic maps from a Riemannian manifold into a Euclidean sphere under the assumption that the image avoids some neighborhood of a half-equator. The proofs combine constructions of strictly convex functions and the regularity theory of quasi-linear elliptic systems. We apply these results to the spherical and Euclidean Bernstein problems for minimal hypersurfaces, obtaining new conditions under which compact minimal hypersurfaces in spheres or complete minimal hypersurfaces in Euclidean spaces are trivial.

Motivation & Objective

• To establish regularity and a-priori estimates for weakly harmonic maps into spheres when their image avoids a neighborhood of a half-equator.
• To extend the classical regularity theory for harmonic maps beyond the open hemisphere condition, achieving a sharper and more general result.
• To apply these regularity results to the spherical and Euclidean Bernstein problems for minimal hypersurfaces.
• To prove that minimal hypersurfaces are trivial (affine planes or equatorial spheres) under topological and geometric constraints on the Gauss map.
• To show that the condition of omitting a neighborhood of a half-equator is optimal, as it prevents the existence of strictly convex functions on larger domains containing closed geodesics.

Proposed method

• Constructing strictly convex functions on compact subsets of the complement of a half-equator in the sphere, tailored to each compact set, even though they are not convex globally.
• Using the composition of harmonic maps with these convex functions to generate weakly subharmonic functions, enabling the application of maximum and Harnack inequalities.
• Applying advanced tools from elliptic PDE theory: Green's test function technique, image shrinking method, telescoping trick, and Harnack inequality methods.
• Employing Moser's Harnack inequality and estimates for Green's functions to control oscillation and energy, independent of the harmonic map's energy.
• Combining the Ruh-Vilms theorem, which links the Gauss map of a minimal hypersurface to harmonic maps into spheres, to translate geometric conditions into analytic ones.
• Using Neumann-Poincaré and volume growth inequalities to verify conditions for Liouville-type theorems on manifolds with nonnegative Ricci curvature or Euclidean volume growth.

Experimental results

Research questions

• RQ1 Can the regularity of weakly harmonic maps into spheres be established under a condition weaker than being contained in an open hemisphere?
• RQ2 Is it possible to derive a-priori estimates for harmonic maps without assuming energy minimality or global energy bounds?
• RQ3 What geometric conditions on the image of the Gauss map ensure that a complete minimal hypersurface in Euclidean space is affine linear?
• RQ4 How does the presence of closed geodesics in the target sphere affect the existence of strictly convex functions and thus the regularity theory?
• RQ5 Can the Bernstein problem for minimal hypersurfaces be resolved under a condition that the Gauss map omits a neighborhood of a half-equator?

Key findings

• Weakly harmonic maps into spheres are regular and satisfy a-priori estimates if their image lies in a compact subset of the complement of a half-equator.
• The construction of strictly convex functions on such sets is possible and essential, even though they are not convex on the entire complement of the half-equator.
• The condition of omitting a neighborhood of a half-equator is optimal, as any larger domain containing a closed geodesic would preclude the existence of strictly convex functions.
• For Riemannian manifolds with nonnegative Ricci curvature, if a weakly harmonic map to the sphere omits a neighborhood of a half-equator, it must be constant.
• For complete minimal hypersurfaces in Euclidean space with Euclidean volume growth and Neumann-Poincaré inequality, if the Gauss map omits a neighborhood of a half-equator, the hypersurface must be an affine linear space.
• The result generalizes and strengthens prior Bernstein-type theorems, including those of Solomon and Wickramasekera, by removing topological assumptions like vanishing first Betti number in some cases.