arXiv research

The paper characterizes equivariant immersions in hyperbolic space.

Computes immersions of $C_2$-projective spaces using K-theory.

The equivariant CR minimal immersions from the round $3$-sphere $S^3$ into the complex projective space $\mathbb CP^n$ have been classified by the third author explicitly (J London Math Soc 68: 223-240, 2003). In this paper, by employing the equivariant condition which implies that the induced metric is left-invariant,…

The study classifies immersed surfaces with knot group Z in simply-connected 4-manifolds.

The paper studies minimal surfaces in complex hyperbolic spaces, showing moduli space connected components.

Proves conditions for minimal surfaces in complex hyperbolic space.

The geometric Hopf invariant of a stable map F is a stable Z_2-equivariant map h(F) such that the stable Z_2-equivariant homotopy class of h(F) is the primary obstruction to F being homotopic to an unstable map. In this paper we express the geometric Hopf invariant of the Umkehr map F of an immersion f:M^m \to N^n in t…

The curvature of almost Fuchsian immersions is concave in their Hopf differentials.

Kahler toric manifolds linked to dually flat spaces via affine isometry.

In this paper, using the framework of equivariant differential geometry, we study proper $SO(p+1) \times SO(q+1)$-invariant biconservative hypersurfaces into the Euclidean space ${\mathbb R}^n$ ($n=p+q+2$) and proper $SO(p+1)$-invariant biconservative hypersurfaces into the Euclidean space ${\mathbb R}^n$ ($n=p+2$). Mo…

In this paper we investigate the relationships between closed AdS 3-manifolds and Higgs bundles. We have a new way to construct AdS structures that allows us to see many of their properties explicitly, for example we can recover the very recent formula by Tholozan for the volumes. We also find applications to the theor…

This paper gives a construction for all minimal immersions $f$ of the Poincaré disc into the complex hyperbolic plane $\mathbb{CH}^2$ which are equivariant with respect to an irreducible representation $ρ$ of a hyperbolic surface group into $PU(2,1)$. We exploit the fact that each such immersion is a twisted conformal …

Minimal surfaces with symmetries exist under certain group actions.

Study immersions of surfaces into SL(2,C) and geodesics space.

For an orbifold, there is a notion of an orbifold embedding, which is more general than the one of sub-orbifolds. We develop several properties of orbifold embeddings. In the case of translation groupoids, we show that such a notion is equivalent to a strong equivariant immersion.

Construct minimal Lagrangian surfaces in complex projective plane via loop group method.

In this article we introduce a definition for the moduli space of equivariant minimal immersions of the Poincaré disc into a non-compact symmetric space, where the equivariance is with respect to representations of the fundamental group of a compact Riemann surface of genus at least two. We then study this moduli space…

Study immersions of surfaces into SL(2,C) with geometric applications.

We consider the eigenfunctions of the Laplace operator $Δ$ on a compact Riemannian manifold of dimension $n$. For $M$ homogeneous with irreducible isotropy representation and for a fixed eigenvalue of $Δ$ we find the average number of common zeros of $n$ eigenfunctions. For this we compute the volume of the image of $M…

Let $S$ be a closed surface of genus at least $2$. For each maximal representation $ρ: π_1(S)\rightarrow\mathsf{Sp}(4,\mathbb{R})$ in one of the $2g-3$ exceptional connected components, we prove there is a unique conformal structure on the surface in which the corresponding equivariant harmonic map to the symmetric spa…

Let $(M,Q)$ be a compact, three dimensional manifold of strictly negative sectional curvature. Let $(Σ,P)$ be a compact, orientable surface of hyperbolic type (i.e. of genus at least two). Let $θ:π_1(Σ,P)\toπ_1(M,Q)$ be a homomorphism. Generalising a recent result of Gallo, Kapovich and Marden concerning necessary and …

The i-th eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of fixed area. Extremal points of these functionals correspond to surfaces admitting minimal isometric immersions into spheres. Recently, critical metrics for the first eigenvalue were classified on tori a…

New invariants derived from link homology for 4-manifolds.

The first author's geometric Hopf invariant of a stable map $F:Σ^{\infty}X \to Σ^{\infty}Y$ is a stable ${\mathbb Z}_2$-equivariant map $h(F):Σ^{\infty}X \to Σ^{\infty}(Y \wedge Y)$ constructed by an explicit difference construction applied to $(F \wedge F)Δ_X - Δ_Y F$. The stable ${\mathbb Z}_2$-equivariant homotopy c…

Let $(S,g_0)$ be a hyperbolic surface, $ρ$ be a Hitchin representation for $PSL(n,\mathbb R)$, and $f$ be the unique $ρ$-equivariant harmonic map from $(\widetilde S, \widetilde g_0)$ to the corresponding symmetric space. We show its energy density satisfies $e(f)\geq 1$ and equality holds at one point only if $e(f)\eq…

An immersion of a smooth $n$-dimensional manifold $M \to \mathbb{R}^q$ is called totally nonparallel if, for every distinct $x, y \in M$, the tangent spaces at $f(x)$ and $f(y)$ contain no parallel lines. Given a manifold $M$, we seek the minimum dimension $TN(M)$ such that there exists a totally nonparallel immersion …

Given a reductive representation $ρ: π_1(S)\rightarrow G$, there exists a $ρ$-equivariant harmonic map $f$ from the universal cover of a fixed Riemann surface $Σ$ to the symmetric space $G/K$ associated to $G$. If the Hopf differential of $f$ vanishes, the harmonic map is then minimal. In this paper, we investigate the…

In 1992, Hitchin used his theory of Higgs bundles to construct an important family of representations of the fundamental group of a closed, oriented surface of genus at least two into the split real form of a complex adjoint simple Lie group. These Hitchin representations comprise a component of the space of conjugacy …

We generalize a well-known existence and uniqueness result for equivariant harmonic maps due to Corlette, Donaldson, and Labourie to a non-compact infinite energy setting and analyze the asymptotic behaviour of the harmonic maps. When the relevant representation is Fuchsian and has hyperbolic monodromy, our constructio…

A classical theorem, mainly due to Aleksandrov and Pogorelov, states that any Riemannian metric on $S^2$ with curvature $K>-1$ is induced on a unique convex surface in $H^3$. A similar result holds with the induced metric replaced by the third fundamental form. We show that the same phenomenon happens with yet another …

Optimizes metrics on surfaces for eigenvalues.

The study finds new constant mean curvature surfaces in curved spaces.

In 2004, Taubes introduced the space of minimal hyperbolic germs with elements consisting of the first and second fundamental form of an equivariant immersed minimal disk in hyperbolic 3-space. Herein, we initiate a further study of this space by studying the behavior of a dynamically defined function which records the…

Study of invariants on manifolds with boundary involving equivariant spectral flow and η-invariants.

Group-equivariant subsampling layers improve CNNs' equivariance.

Study of equivariant ribbon concordance using Khovanov homology.

Defines a new homomorphism for strongly invertible knots, proving equivariant algebraic concordance.

Defines an equivariant Ruelle dynamical zeta function for flows on manifolds.

Study of equivariant movie moves for involutive links.

We define the equivariant holonomy of an invariant connection on a principal U(1)-bundle. The properties of the ordinary holonomy are generalized to the equivariant setting. In particular, equivariant U(1)-bundles with connection are shown to be classified by its equivariant holonomy modulo isomorphisms. We also show t…

A bound on knot unknotting using equivariant signature.

New proof for 4D symplectic manifolds: equivariant cohomology determines diffeotype.

The paper develops methods for calculating equivariant homology from Morse functions.

Efficiently samples and learns densities with symmetries using equivariant methods.

The study constructs immersions with controlled curvatures between manifolds and identifies obstacles.

Proves bijection between smooth conformal immersions and immersions.

Study shows non-equivariant and equivariant non-orientable 4-genus of periodic knots can differ.

An immersion of a compact manifold is tight if it admits the minimal total absolute curvature over all immersions of the manifold. A prominent result in the study of minimal total absolute curvature immersions is the theorem of Chern and Lashof, which characterizes minimal total absolute curvature immersions, and tight…