Abstract In the joint work of the author with Da Lio, F., and Rivière, T., Schlagenhauf, D.: (2025) we studied the stability of the Morse index for Sacks-Uhlenbeck sequences into spheres as $$p\searrow 2$$ p ↘ 2 . These are critical points of the energy $$E_p(u) {:}{=}\int _\Sigma \left( 1+\left| \nabla u\right| ^2\right) ^{p/2} \ dvol_\Sigma ,$$ E p ( u ) : = ∫ Σ 1 + ∇ u 2 p / 2 d v o l Σ , where $$u:\Sigma \rightarrow S^n$$ u : Σ → S n is a map from a closed Riemannian surface $$\Sigma $$ Σ into a sphere $$ S^n$$ S n . In this paper we extend the results found in Da Lio, F., Rivière, T., Schlagenhauf, D.: (2025) to the case of Sacks-Uhlenbeck sequences into homogeneous spaces, by incorporating the strategy introduced in Bayer, C., and Roberts, A.: (2025) . In the spirit of Da Lio, F., Rivière, T., Schlagenhauf, D.: (2025), we show in this setting the upper semicontinuity of the Morse index plus nullity and an improved pointwise estimate of the gradient in the neck regions around blow up points.

Abstract We prove a suite of results classifying holomorphic maps between configuration spaces of Riemann surfaces; we consider both the ordered and unordered setting as well as the cases of genus zero, one, and at least two. We give a complete classification of all holomorphic maps $$\operatorname {Conf}_n(\mathbb {C})\rightarrow \operatorname {Conf}_m(\mathbb {C})$$ Conf n ( C ) → Conf m ( C ) provided that $$n\ge 5$$ n ≥ 5 and $$m\le 2n$$ m ≤ 2 n extending the Tameness Theorem of Lin, which is the case $$m = n$$ m = n . We also give a complete classification of holomorphic maps between ordered configuration spaces of Riemann surfaces of genus at most one (answering a question of Farb), and show that the higher genus setting is closely linked to the still-mysterious “effective de Franchis problem”. The main technical theme of the paper is that holomorphicity allows one to promote group-theoretic rigidity results to the space level.

Abstract We prove that over an algebraically closed field of characteristic $$p>0$$ p > 0 there are exactly, up to isomorphism, n infinitesimal commutative unipotent k -group schemes of order $$p^n$$ p n with one-dimensional Lie algebra, and we explicitly describe them. We consequently obtain an explicit description of all infinitesimal subgroup schemes of any supersingular elliptic curve over an algebraically closed field, recovering all their $$p^n$$ p n -torsions as well. Finally, we use these results to answer a question of Brion on rational actions of infinitesimal commutative unipotent group schemes on curves.