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For a harmonic diffeomorphism between the Poincaré disks, Wan [J. Differential Geom. 35 (1992), pp. 643–657] showed the equivalence between the boundedness of the Hopf differential and the quasi-conformality. In this paper, we will generalize this result from quadratic differentials to r-differentials. We study the relationship between bounded holomorphic r-differentials/(r-1)-differential and the induced curvature of the associated harmonic maps from the unit disk to the symmetric space SL(r,\mathbb R)/SO(r) arising from cyclic/subcyclic Higgs bundles. Also, we show the equivalence between the boundedness of holomorphic differentials and having a negative upper bound of the induced curvature on hyperbolic affine spheres in \mathbb {R}^3, maximal surfaces in \mathbb {H}^{2,n} and J-holomorphic curves in \mathbb {H}^{4,2}. Benoist-Hulin and Labourie-Toulisse have previously obtained some of these equivalences using different methods. [ABSTRACT FROM AUTHOR]

The geproci property is a recent development in the world of geometry. We call a set of points Z\subseteq \mathbb {P}_k^3 an (a,b)-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point P to a plane is a complete intersection of curves of degrees a\leq b. Nondegenerate examples known as grids have been known since 2011. Nondegenerate nongrids were first described in 2018, working in characteristic 0. Almost all of these new examples are of a special kind called half grids. In this paper, based partly on the author's thesis, we use a feature of geometry in positive characteristic to give new methods of producing geproci half grids and non-half grids. [ABSTRACT FROM AUTHOR]

In this paper, we study, for functionals having a minimax geometry on a constraint, the existence of bounded Palais-Smale sequences carrying Morse index type information. [ABSTRACT FROM AUTHOR]

We prove an equivalence between the superpotential defined via tropical geometry and Lagrangian Floer theory for special Lagrangian torus fibres in del Pezzo surfaces constructed by Collins-Jacob-Lin [Duke Math. J. 170 (2021), pp. 1291–1375]. We also include some explicit calculations for the projective plane, which confirm some folklore conjectures in this case. [ABSTRACT FROM AUTHOR]

In 1990, Schnyder used a 3-spanning-tree decomposition of a simple triangulation, now known as the Schnyder wood , to give a fundamental grid-embedding algorithm for planar maps. In the framework of mating of trees, a uniformly sampled Schnyder-wood-decorated triangulation can produce a triple of random walks. We show that these three walks converge in the scaling limit to three Brownian motions produced in the mating-of-trees framework by Liouville quantum gravity (LQG) with parameter 1, decorated with a triple of SLE_{16}'s curves. These three SLE_{16}'s curves are coupled such that the angle difference between them is 2\pi /3 in imaginary geometry. Our convergence result provides a description of the continuum limit of Schnyder's embedding algorithm via LQG and SLE. [ABSTRACT FROM AUTHOR]

A locally conformally Kähler (LCK) manifold is a complex manifold M which has a Kähler structure on its cover, such that the deck transform group acts on it by homotheties. Assume that the Kähler form is exact on the minimal Kähler cover of M. We prove that any bimeromorphic map M'\rightarrow M is in fact holomorphic; in other words, M has a unique minimal model. This can be applied to a wide class of LCK manifolds, such as the Hopf manifolds, their complex submanifolds and to OT manifolds. [ABSTRACT FROM AUTHOR]

We prove that the Ricci flow for complete metrics with bounded geometry depends continuously on initial conditions for finite time with no loss of regularity. This relies on recent work of Bahuaud, Guenther, Isenberg and Mazzeo where sectoriality for the generator of the Ricci-DeTurck flow is proved. We use this to prove that for initial metrics sufficiently close in Hölder norm to a rotationally symmetric asymptotically hyperbolic metric and satisfying a simple curvature condition, but a priori distant from the hyperbolic metric, Ricci flow converges to the hyperbolic metric. [ABSTRACT FROM AUTHOR]

We study compactifications of the moduli space of unordered points in the plane via variation of GIT-quotients of their corresponding Hilbert scheme. Our VGIT considers linearizations outside the ample cone and within the movable cone. For that purpose, we use the description of the Hilbert scheme as a Mori dream space, and the moduli interpretation of its birational models via Bridgeland stability. We determine the GIT walls associated with curvilinear zero-dimensional schemes, collinear points, and schemes supported on a smooth conic. For seven points, we study a compactification associated with an extremal ray of the movable cone, where stability behaves very differently from the Chow quotient. Lastly, a complete description for five points is given. [ABSTRACT FROM AUTHOR]