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The boundary value problem for triholomorphic maps
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 20 November 2025, pp. 1-29
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- Article
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We introduce a natural boundary value problem for a triholomorphic map
$u$ from a compact almost hyper-Hermitian manifold 
$M$ with smooth boundary 
$\partial M$ into a closed hyperKähler manifold 
$N$ with free boundary 
$u(\partial M)\subset \Gamma$ lying on some geometrically natural closed supporting submanifold 
$\Gamma\subset N$, called tri-isotropic submanifold. We establish partial regularity theory and energy quantization result in this boundary setting under some additional assumption on the 
$W^{2,1}$ norm of the weakly converging sequences.
