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Inkang Kim

Publications and source records attributed to Inkang Kim.

At least 19 recordsLinked to original sources
arXiv2026

Quasi-convexity of energy functions along Teichm\"uller geodesics

Hyperbolic length functions are among the most fundamental ones on Teichm\"uller space, and they are quasi-convex along Teichm\"uller geodesics. In this paper, we investigate the same question for energy functions of harmonic maps in two natural settings, which may be viewed as nonlinear and higher-dimensional analogs of the length functions. For a fixed domain and a varying hyperbolic target, we prove the energy is quasi-convex along Teichm\"uller geodesics under a filling hypothesis. Furthermore, we generalize Masur's result on asymptotic growth of the length function along the Teichm\"uller geodesic determined by a Jenkins-Strebel differential to the energy functions. We also prove the quasi-convexity for covering maps between closed hyperbolic surfaces with fixed target and varying domains. We derive first and second variation formulas of energy functions along Teichm\"uller geodesics and explain why the natural global statement is quasi-convexity rather than genuine convexity.

math.DG↗
arXiv2025

Topological components of surface group representations and signature

We study the topological components of the surface group representations into $\mathrm{SL}(2,\mathbb{R})$ and $\mathrm{PSL}(2,\mathbb{R})$. Utilizing the signature formula established in [14], we determine the number of connected components of the representation spaces with boundary elliptic, hyperbolic, and parabolic holonomies.

math.GT↗
arXiv2025

Positivity of simplicial volume for closed nonpositively curved four-manifolds with nonzero Euler characteristic

In this paper, by employing the Gauss-Bonnet theorem for Riemannian simplices due to Allendoerfer and Weil, we show that if a closed nonpositively curved $4$-manifold has nonzero Euler characteristic, then its simplicial volume is necessarily positive. This result partially resolves conjectures posed by Connell-Ruan-Wang and Gromov concerning the relationship between the simplicial volume and the Euler characteristic for four-dimensional manifolds. As an application, we show that if a closed nonpositively curved $4$-manifold has negative Ricci curvature, then its simplicial volume is positive, thereby confirming in dimension four another conjecture of Gromov on the positivity of simplicial volume.

math.GT↗
arXiv2022

Curvature of the total space of a Griffiths negative vector bundle and quasi-Fuchsian space

For a holomorphic vector bundle $E$ over a Hermitian manifold $M$ there are two important notions of curvature positivity, the Griffiths positivity and Nakano positivity. We study the consequence of these positivities and the relevant estimates. If $E$ is Griffiths negative over K\"ahler manifold, then there is a K\"ahler metric on its total space $E$, and we calculate the curvature and prove the non-positivity of the curvature along the tautological direction. The Nakano positivity can be formulated as a positivity for the Nakano curvature operator and we give estimate the Nakano curvature operator associated with a Nakano positive direct image bundle. As applications we construct a mapping class group invariant K\"ahler metric on the quasi-Fuchsian space QF$(S)$, which extends the Weil-Petersson metric on the Teichm\"uller space $\mathcal{T}(S)\subset {\rm QF}(S)$, and we obtain estimates for the Nakano curvature operator for the dual Weil-Petersson metric on the holomorphic cotangent bundle of Teichm\"uller space.

math.DG↗
arXiv2022

Signature for flat unitary bundles over surfaces with boundary

This paper deals with the representations of the fundamental groups of compact surfaces with boundary into classical simple Lie groups of Hermitian type. We relate work on the signature of the associated local systems of Atiyah-Patodi-Singer, to Burger-Iozzi-Wienhard's Toledo invariant. To measure the difference, we extend Atiyah-Patodi-Singer's rho invariant, initially defined on $\mathrm{U}(p)$, to discontinuous class functions, first on $\mathrm{U}(p,q)$, and then on other classical groups via embeddings into $\mathrm{U}(p,q)$. In this way, we present three different invariants -- signature, Toledo and rho invariant -- in a unifying way, which is a version of the classical signature formula of Atiyah-Patodi-Singer for manifolds with boundary.

math.GT↗
arXiv2021

Signature, Toledo invariant and surface group representations in the real symplectic group

In this paper, by using Atiyah-Patodi-Singer index theorem, we obtain a formula for the signature of a flat symplectic vector bundle over a surface with boundary, which is related to the Toledo invariant of a surface group representation in the real symplectic group and the Rho invariant on the boundary. As an application, we obtain a Milnor-Wood type inequality for the signature. In particular, we give a new proof of the Milnor-Wood inequality for the Toledo invariant in the case of closed surfaces and obtain some modified inequalities for the surface with boundary.

math.GT↗
arXiv2021

Scalar curvature, mean curvature and harmonic maps to the circle

We study harmonic maps from a 3-manifold with boundary to $\mathbb{S}^1$ and prove a special case of dihedral rigidity of three dimensional cubes whose dihedral angles are $\pi / 2$. Furthermore we give some applications to mapping torus hyperbolic 3-manifolds.

math.DG↗
arXiv2020

Harmonic maps between surfaces homotopic to a (branched) covering map

In the paper, we consider the harmonic maps between surfaces $\Sigma$ and $S$ in the homotopy class of a (branched) covering map $u_0$. We prove the uniqueness of critical points of energy function and the injectivity of Hopf differential if $u_0$ is a covering map. On the other hand, if $u_0$ is a branched covering, we show that the uniqueness of critical points fails if $u_0$ is a non-simple branched covering, and prove the injectivity of Hopf differential $\Phi:\mc{T}(S)\to \op{QD}(\Sigma,g)$ when $g=[u_0^* h]$ for some hyperbolic metric $h$ on $S$.

math.DG↗
arXiv2019

Convexity of energy function associated to the harmonic maps between surfaces

For a fixed smooth map $u_0$ between two Riemann surfaces $\Sigma$ and $S$ with non-zero degree, we consider the energy function on Teichm\"uller space $\mc{T}$ of $\Sigma$ that assigns to a complex structure $t\in \mc{T}$ on $\Sigma$ the energy of the harmonic map $u_t:\Sigma_t:=(\Sigma,t) \to S$ homotopic to $u_0$. We prove that the energy function is convex at its critical points. If $t_0\in\mc{T}$ is a critical point such that $du_{t_0}$ is never zero, then the energy function is strictly convex at this point. As an application, in the case that $u_0$ is a covering map, we prove that there exists a unique critical point $t_0\in \mc{T}$ minimizing the energy function. Moreover, the energy density satisfies $\frac{1}{2}|du|^2(t_0)\equiv 1$ and the Hessian of the energy function is positive definite at this point.

math.DG↗
arXiv2019

Systole on locally symmetric spaces

Here we survey on the growth of systoles of arithmetic locally symmetric spaces under the congruence covering and give simple proofs for the best possible constants of Gromov for several important classes of symmetric spaces.

math.DG↗
arXiv2019

New K\"ahler metric on quasifuchsian space and its curvature properties

Let $QF(S)$ be the quasifuchsian space of a closed surface $S$ of genus $g\geq 2$. We construct a new mapping class group invariant K\"ahler metric on $QF(S)$. It is an extension of the Weil-Petersson metric onthe Teichm\"uller space $\mathcal T(S)\subset QF(S)$. We also calculate its curvature and prove some negativity for the curvature along the tautological directions.

math.GT↗
arXiv2019

Plurisuperharmonicity of reciprocal energy function on Teichmuller space and Weil-Petersson metrics

We consider harmonic maps$u(z): \mathcal{X}_z\to N$ in a fixed homotopy class from Riemann surfaces $\mathcal{X}_z$ of genus $g\geq 2$ varying in the Teichm\"u{}ller space $\mathcal T$ to a Riemannian manifold $N$ with non-positive Hermitian sectional curvature. The energy function $E(z)=E(u(z))$ can be viewed as a function on $\mathcal T$ and we study its first and the second variations. We prove that the reciprocal energy function $E(z)^{-1}$ is plurisuperharmonic on Teichm\"uller space. We also obtain the (strict) plurisubharmonicity of $\log E(z)$ and $E(z)$. As an application, we get the following relationship between the second variation of logarithmic energy function and the Weil-Petersson metric if the harmonic map $u(z)$ is holomorphic or anti-holomorphic and totally geodesic, i.e., $$ \sqrt{-1}\p\b{\p}\log E(z)=\frac{\omega_{WP}}{2\pi(g-1)}. $$ We consider also the energy function $E(z)$ associated to the harmonic maps from a fixed compact K\"ahler manifold $M$ to Riemann surfaces ${\mathcal{X}_z\}_{z\in\mathcal{T}}$ in a fixed homotopy class. If $u(z)$ is holomorphic or anti-holomorphic, then the above equation is also proved.

math.DG↗
arXiv2018

On the Markus conjecture in convex case

In this paper, we show that any convex affine domain with a nonempty limit sets on the boundary under the action of the identity component of the automorphism group cannot cover a compact affine manifold with a parallel volume, which is a positive answer to the Markus conjecture for convex case. Consequently, we show that the Markus conjecture is true for convex affine manifolds of dimension $\leq 5$.

math.GT↗