arXiv subjects

Anton Iliashenko

Publications and source records attributed to Anton Iliashenko.

4 recordsLinked to original sources

arXiv2026

Montel's theorem and tautness in calibrated geometry

We relate the hyperbolicity of a calibrated manifold $(X, ϕ)$ to the analytic properties of the space of Smith immersions $\mathrm{SmIm}(B^k, X)$ from the Poincare $k$-ball into $X$. In particular, we establish the following calibrated analogue of a theorem of Royden: if $X$ is $ϕ$-replete, then $R_ϕ$- and $K_ϕ$-hyperbolicity coincide, and either implies the equicontinuity of $\mathrm{SmIm}(B^k, X)$ with respect to the $ϕ$-distance. This yields a Montel theorem for compact $ϕ$-replete calibrated manifolds as an immediate corollary. Our primary technical tool is a new Schwarz lemma for Smith immersions from $B^k$ into $X$, which is of independent interest. In a similar spirit, we also prove a calibrated analogue of Kiernan's theorem to the effect that the $K_ϕ$-hyperbolicity of $X$ is almost equivalent to $\mathrm{SmIm}(B^k, X)$ being a normal family. Finally, we prove that bounded domains in flat euclidean space are $R_ϕ$-hyperbolic for any calibration $ϕ$, and we investigate the hyperbolicity of products and discrete quotients.

math.DG↗

arXiv2025

Hyperbolicity and Schwarz Lemmas in Calibrated Geometry

This paper has two main objectives. First, for an arbitrary calibrated manifold $(X,ϕ)$, we define notions of $R_ϕ$-hyperbolicity and $ϕ$-hyperbolicity, which respectively generalize the notions of Kobayashi and Brody hyperbolicity from complex geometry. To make sense of the former, we introduce the "KR $ϕ$-metric," a decreasing Finsler pseudo-metric that specializes to the Kobayashi-Royden pseudo-metric in the Kahler case. We prove that $R_ϕ$-hyperbolicity implies $ϕ$-hyperbolicity, and give examples showing that the converse fails in general. Moreover, for constant-coefficient, inner Mobius rigid calibrations $ϕ$ in $\mathbb{R}^n$, we completely characterize those domains that are $ϕ$-hyperbolic. Second, we derive a Schwarz lemma for Smith immersions (a.k.a. conformal $ϕ$-curves) into an arbitrary calibrated manifold $(X, ϕ)$, thereby extending the Schwarz lemma for holomorphic curves into Kahler manifolds. The relevant Bochner formula features the "$ϕ$-sectional curvature," a new notion that includes both the scalar and holomorphic sectional curvatures as special cases. As an application, we prove that calibrated geometries with $ϕ$-sectional curvature bounded above by a negative constant are $R_ϕ$-hyperbolic, generalizing the corresponding result from complex geometry. As another application, we calculate the KR $ϕ$-metric of real, complex, and quaternionic hyperbolic spaces equipped with their natural calibrations.

math.DG↗

arXiv2025

A special class of $k$-harmonic maps inducing calibrated fibrations

We consider two special classes of $k$-harmonic maps between Riemannian manifolds which are related to calibrated geometry, satisfying a first order fully nonlinear PDE. The first is a special type of weakly conformal map $u \colon (L^k, g) \to (M^n, h)$ where $k \leq n$ and $α$ is a calibration $k$-form on $M$. Away from the critical set, the image is an $α$-calibrated submanifold of $M$. These were previously studied by Cheng-Karigiannis-Madnick when $α$ was associated to a vector cross product, but we clarify that such a restriction is unnecessary. The second, which is new, is a special type of weakly horizontally conformal map $u \colon (M^n, h) \to (L^k, g)$ where $n \geq k$ and $α$ is a calibration $(n-k)$-form on $M$. Away from the critical set, the fibres $u^{-1} \{ u(x) \}$ are $α$-calibrated submanifolds of $M$. We also review some previously established analytic results for the first class; we exhibit some explicit noncompact examples of the second class, where $(M, h)$ are the Bryant-Salamon manifolds with exceptional holonomy; we remark on the relevance of this new PDE to the Strominger-Yau-Zaslow conjecture for mirror symmetry in terms of special Lagrangian fibrations and to the $\mathrm{G}_2$ version by Gukov-Yau-Zaslow in terms of coassociative fibrations; and we present several open questions for future study.

math.DG↗

arXiv2024

Betti numbers of nearly $G_2$ and nearly Kähler manifolds with Weyl curvature bounds

In this paper we use the Weitzenböck formulas to get information about the Betti numbers of compact nearly $G_2$ and compact nearly Kähler $6$-manifolds. First, we establish estimates on two curvature-type self adjoint operators on particular spaces assuming bounds on the sectional curvature. Then using the Weitzenböck formulas on harmonic forms, we get results of the form: if certain lower bounds hold for these curvature operators then certain Betti numbers are zero. Finally, we combine both steps above to get sufficient conditions of vanishing of certain Betti numbers based on the bounds on the sectional curvature.

math.DG↗