Q-valued functions revisited

Abstract

In this note we revisit Almgren's theory of Q-valued functions, that are functions taking values in the space of unordered Q-tuples of points in R^n. In particular: 1) we give shorter versions of Almgren's proofs of the existence of Dir-minimizing Q-valued functions, of their Hoelder regularity and of the dimension estimate of their singular set; 2) we propose an alternative intrinsic approach to these results, not relying on Almgren's biLipschitz embedding; 3) we improve upon the estimate of the singular set of planar Dir-minimizing functions by showing that it consists of isolated points.

84 pages, 3 figures; changes: few typos corrected, reformatted for submission

References in corpus (6)

• Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions
• Higher integrability and approximation of minimal currents
• A regularity theory for multiple-valued Dirichlet minimizing maps
• An Energy Reducing Flow for Multiple-Valued Functions
• Analysis on Metric Space Q
• A Theorem on Frequency Function for Multiple-Valued Dirichlet Minimizing Functions

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• Semi-calibrated 2-currents are pseudo holomorphic, with applications
• Higher integrability and approximation of minimal currents
• Selections of bounded variation for roots of smooth polynomials
• Existence of $p$ harmonic multiple valued maps into a separable Hilbert space
• Multiple valued sections of vector bundles: the reparametrization theorem for $Q$-valued functions revisited
• Kirszbraun's extension theorem fails for Almgren's multiple valued functions
• Fine Structure of Singularities in Area-Minimizing Currents Mod$(q)$
• Boundary regularity for anisotropic minimal Lipschitz graphs
• Multiple valued Jacobi fields
• Optimal Sobolev regularity of roots of polynomials
• Partial Hölder continuity for Q-valued energy minimizing maps
• Complex varieties and higher integrability of Dir-minimizing Q-valued functions
• Sobolev Lifting over Invariants
• Integral decompositions of varifolds
• Fractional Parabolic Theory as a High-Dimensional Limit of Fractional Elliptic Theory