Perturbed cone theorems for proper harmonic maps
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Abstract
Abstract Inspired by the halfspace theorem for minimal surfaces in upper R cubed $\mathbb {R}^3$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mstyle mathvariant="double-struck"> <mml:msup> <mml:mi>R</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:mstyle> </mml:math> of Hoffman–Meeks, the halfspace theorem of Rodriguez–Rosenberg, and the classical cone theorem of Omori in upper R Superscript n $\mathbb {R}^n$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mstyle mathvariant="double-struck"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mstyle> </mml:math> , we derive new non-existence results for proper harmonic maps into perturbed cones in upper R Superscript n $\mathbb {R}^n$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mstyle mathvariant="double-struck"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mstyle> </mml:math> , horospheres in upper H Superscript n $\mathbb {H}^n$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mstyle mathvariant="double-struck"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mstyle> </mml:math> , culminating in a generalization of Omori’s theorem in arbitrary Riemannian manifolds. The technical tool proved here extends the foliated Sampson’s maximum principle, initially developed in the first author’s Ph.D. thesis, to a non-compact setting.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.009 | 0.001 |
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