Abstract We study the positive Hermitian curvature flow for left-invariant metrics on 2-step nilpotent Lie groups G with a left-invariant complex structure J . We describe the long-time behavior of the flow under the assumption that $$J[\mathfrak {g}, \mathfrak {g}]$$ J [ g , g ] is contained in the center of $$\mathfrak {g}$$ g . We show that under our assumption the flow $$g_{t}$$ g t exists for all positive t and $$(G,(1+t)^{-1}g_{t})$$ ( G , ( 1 + t ) - 1 g t ) converges, in the Cheeger-Gromov topology, to a 2-step nilpotent Lie group with a non flat semi-algebraic soliton. Moreover, we prove that, in our class of Lie groups, there exists at most one semi-algebraic soliton solution, up to homothety. Similar results were proved by M. Pujia and J. Stanfield for nilpotent complex Lie groups [21, 24]. In the last part of the paper we study the Hermitian curvature flow for the same class of Lie groups.

Abstract We prove that if a pair of Kähler classes is J -nef, the J -flow on a compact Kähler surface converges to a weak solution of the Monge-Ampère equation in the sense of currents. We also establish the same convergence behavior for the deformed Hermitian-Yang-Mills flow. The method is based on a property of a limit of viscosity subsolutions.

Abstract We show that for any Riemannian foliation with a simply connected and negatively curved leaf space the normal exponential map of a leaf is a diffeomorphism. As an application, if the leaves are furthermore minimal submanifolds, we give a sharp estimate for the bottom of the spectrum of such a Riemannian manifold. Our proof of the spectral estimate also yields an estimate for the bottom of the spectrum of the horizontal Laplacian.

Abstract For a small polarised deformation of a constant scalar curvature Kähler manifold, under some cohomological vanishing conditions, we prove that K-polystability along nearby polarisations implies the existence of a constant scalar curvature Kähler metric. In this setting, we reduce K-polystability to the computation of the classical Futaki invariant on the cscK degeneration. Our result holds on specific families and provides local wall-crossing phenomena for the moduli of cscK manifolds when the polarisation varies.

For a Gelfand pair (G, K) with G a Lie group of polynomial growth and K a compact subgroup, the Schwartz correspondence states that the spherical transform maps the bi-K-invariant Schwartz space S(K\\G/K)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {S}}}(K\\backslash G/K)$$\\end{document} isomorphically onto the space S(ΣD)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {S}}}(\\Sigma _{{\\mathcal {D}}})$$\\end{document}, where ΣD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma _{{\\mathcal {D}}}$$\\end{document} is an embedded copy of the Gelfand spectrum in Rℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {R}}}^\\ell $$\\end{document}, canonically associated to a generating system D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {D}}}$$\\end{document} of G-invariant differential operators on G/K, and S(ΣD)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {S}}}(\\Sigma _{{\\mathcal {D}}})$$\\end{document} consists of restrictions to ΣD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma _{{\\mathcal {D}}}$$\\end{document} of Schwartz functions on Rℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {R}}}^\\ell $$\\end{document}. Schwartz correspondence is known to hold for a large variety of Gelfand pairs of polynomial growth. In this paper we prove that it holds for the strong Gelfand pair (Mn,SOn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(M_n,SO_n)$$\\end{document} with n=3,4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n=3,4$$\\end{document}. The rather trivial case n=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n=2$$\\end{document} is included in previous work by the same authors.