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Hojoo Lee

Publications and source records attributed to Hojoo Lee.

At least 19 recordsLinked to original sources
arXiv2023

A duality for prescribed mean curvature graphs in Riemannian and Lorentzian Killing submersions

We develop a conformal duality for spacelike graphs in Riemannian and Lorentzian three-manifolds that admit a Riemannian submersion over a Riemannian surface whose fibers are the integral curves of a Killing vector field, which is timelike in the Lorentzian case. The duality swaps mean curvature and bundle curvature and sends the length of the Killing vector field to its reciprocal while keeping invariant the base surface. We obtain two consequences of this result. On the one hand, we find entire graphs in Lorentz-Minkowski space $\mathbb{L}^3$ with prescribed mean curvature a bounded function $H\in C^\infty(\mathbb{R}^2)$ with bounded gradient. On the other hand, we obtain conditions for existence and non existence of entire graphs which are related to a notion of critical mean curvature.

math.DG↗
arXiv2022

Marginally trapped surfaces in ${\mathbb{L}}^{4}$ and three Weierstrass representations

We construct new integrable systems to present Weierstrass type representations for spacelike surfaces whose mean curvature vector $\mathbf{H}$ satisfies the null condition $\langle \mathbf{H}, \mathbf{H} \rangle=0$ in the four dimensional Lorentz-Minkowski space ${\mathbb{L}}^{4}$. Our new Weierstrass presentations extend simultaneously classical Weierstrass representations (of the first kind and the second kind) for maximal surfaces in ${\mathbb{L}}^{3}$ and minimal surfaces in ${\mathbb{R}}^{3}$. We solve a linear partial differental equation to construct explicit examples of marginally trapped surfaces with nowhere vanishing mean curvature vector.

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arXiv2022

Length-minimizing level curves via calibrations

We present an elementary criterion to show the length-minimizing property of geodesics for a large class of conformal metrics. In particular, we prove the length-minimizing property of level curves of harmonic functions and the length-minimizing property of a family of the conic sections with the eccentricity $\varepsilon$ in the upper half plane endowed with the conformal metric $ \left( {\varepsilon}^{2} + \frac{1}{\;{y^2} \;} \right) \left(dx^{2} + dy^{2} \right)$.

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arXiv2019

Uniqueness of minimal surfaces, Jacobi fields, and flat structures

Inspired by the Finn-Osserman (1964), Chern (1969), do Carmo-Peng (1979) proofs of the Bernstein theorem, which characterizes flat planes as the only entire minimal graphs, we prove a new rigidity theorem for associate families connecting the doubly periodic Scherk graphs and the singly periodic Scherk towers. Our characterization of Scherk's surfaces discovers a new idea from the original Finn-Osserman curvature estimate. Combining two generically independent flat structures introduced by Chern and Ricci, we shall construct geometric harmonic functions on minimal surfaces, and establish that periodic minimal surfaces admit fresh uniqueness results.

math.DG↗
arXiv2018

Two flat structures on minimal surfaces

In this expository article, we illustrate how two independent flat structures on minimal surfaces induce a harmonic function, which captures the uniqueness of Enneper's surface.

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arXiv2017

A survey of closed self-shrinkers with symmetry

In this paper, we survey known results on closed self-shrinkers for mean curvature flow and discuss techniques used in recent constructions of closed self-shrinkers with classical rotational symmetry. We also propose new existence and uniqueness problems for closed self-shrinkers with bi-rotational symmetry and provide numerical evidence for the existence of new examples.

math.DG↗
arXiv2017

Minimal surface system in Euclidean four-space

Generalizing the Cauchy-Riemann equations, we construct the Osserman system of the first order for a pair $\left(f(x, y), g(x,y) \right)$ of two ${\mathbb{R}}$-valued functions on the domain $Ω\subset {\mathbb{R}}^{2}$. The graph $\left\{\, \left(x, y, f(x, y), g(x,y) \right) \in {\mathbb{R}}^{4} \, \vert \, (x,y) \in Ω\, \right\}$ becomes a minimal surface in ${\mathbb{R}}^{4}$, whose generalized Gauss map lies on the intersection of a hyperplane of the complex projective space ${\mathbb{CP}}^{3}$ and the complex cone ${z_1}^{2}+{z_2}^{2}+{z_3}^{2}+{z_4}^{2}=0$. We present two applications of the Lagrangian potential on minimal graphs in ${\mathbb{R}}^{3}$. First, we deform a minimal graph $Σ_{0}$ in ${\mathbb{R}}^{3}$ to the one parameter family of the two dimensional minimal graph $Σ_λ$ in ${\mathbb{R}}^{4}$ with the invariance of the metric ${\left(\det{ \left( {\mathbf{g}}_{Σ_λ} \right) }\right)}^{- \frac{1}{2}} {\mathbf{g}}_{Σ_λ}$. Second, we construct the three dimensional special Lagrangian graphs in ${\mathbb{R}}^{6}={\mathbb{C}}^{3}$.

math.DG↗
arXiv2017

Minimal surfaces in ${\mathbb{R}}^{4}$ foliated by conic sections and parabolic rotations of holomorphic null curves in ${\mathbb{C}}^{4}$

Using the complex parabolic rotations of holomorphic null curves in ${\mathbb{C}}^{4}$, we transform minimal surfaces in Euclidean space ${\mathbb{R}}^{3} \subset {\mathbb{R}}^{4}$ to a family of degenerate minimal surfaces in Euclidean space ${\mathbb{R}}^{4}$. Applying our deformation to holomorphic null curves in ${\mathbb{C}}^{3} \subset {\mathbb{C}}^{4}$ induced by helicoids in ${\mathbb{R}}^{3}$, we discover new minimal surfaces in ${\mathbb{R}}^{4}$ foliated by conic sections with eccentricity grater than $1$: hyperbolas or straight lines. Applying our deformation to holomorphic null curves in ${\mathbb{C}}^{3}$ induced by catenoids in ${\mathbb{R}}^{3}$, we can rediscover the Hoffman-Osserman catenoids in ${\mathbb{R}}^{4}$ foliated by conic sections with eccentricity smaller than $1$: ellipses or circles. We prove the existence of minimal surfaces in ${\mathbb{R}}^{4}$ foliated by ellipses, which converge to circles at infinity. We construct minimal surfaces in ${\mathbb{R}}^{4}$ foliated by parabolas: conic sections which have eccentricity $1$.

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arXiv2017

The uniqueness of the Enneper surfaces and Chern-Ricci functions on minimal surfaces

We construct the first and second Chern-Ricci functions on negatively curved minimal surfaces in ${\mathbb{R}}^{3}$ using Gauss curvature and angle functions, and establish that they become harmonic functions on the minimal surfaces. We prove that a minimal surface has constant first Chern-Ricci function if and only if it is Enneper's surface. We explicitly determine the moduli space of minimal surfaces having constant second Chern-Ricci function, which contains catenoids, helicoids, and their associate families.

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arXiv2016

Weighted Hsiung-Minkowski formulas and rigidity of umbilical hypersurfaces

We use the weighted Hsiung-Minkowski integral formulas and Brendle's inequality to show new rigidity results. First, we prove Alexandrov type results for closed embedded hypersurfaces with radially symmetric higher order mean curvature in a large class of Riemannian warped product manifolds, including the Schwarzschild and Reissner-Nordström spaces, where the Alexandrov reflection principle is not available. Second, we prove that, in Euclidean space, the only closed immersed self-expanding solitons to the weighted generalized inverse curvature flow of codimension one are round hyperspheres.

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arXiv2015

Generalizations of the Choe-Hoppe helicoid and Clifford cones in Euclidean space

Our goal is to generalize the Choe-Hoppe helicoid and Clifford cones in Euclidean space. By sweeping out $L$ indpendent Clifford cones in ${\mathbb{R}}^{2N+2}$ via the multi-screw motion, we construct minimal submanifolds in ${\mathbb{R}}^{L(2N+2)+1}$. Also, we sweep out the $L$-rays Clifford cone (introduced in Section 2.3) in ${\mathbb{R}}^{L(2N+2)}$ to construct minimal submanifolds in ${\mathbb{R}}^{L(2N+2)+1}$. Our minimal submanifolds unify various interesting examples: Choe-Hoppe's helicoid of codimension one, cone over Lawson's ruled minimal surfaces in ${\mathbb{S}}^{3}$, Barbosa-Dajczer-Jorge's ruled submanifolds, and Harvey-Lawson's volume-minimizing twisted normal cone over the Clifford torus.

math.DG↗
arXiv2015

Solitons for the inverse mean curvature flow

We investigate self-similar solutions to the inverse mean curvature flow in Euclidean space. In the case of one dimensional planar solitons, we explicitly classify all homothetic solitons and translators. Generalizing Andrews' theorem that circles are the only compact homothetic planar solitons, we apply the Hsiung-Minkowski integral formula to prove the rigidity of the hypersphere in the class of compact expanders of codimension one. We also establish that the moduli space of compact expanding surfaces of codimension two is big. Finally, we update the list of Huisken-Ilmanen's rotational expanders by constructing new examples of complete expanders with rotational symmetry, including topological hypercylinders, called infinite bottles, that interpolate between two concentric round hypercylinders.

math.DG↗
arXiv2013

Generalized Calabi correspondence and complete spacelike surfaces

We construct a twin correspondence between graphs with prescribed mean curvature in three-dimensional Riemannian Killing submersions and spacelike graphs with prescribed mean curvature in three-dimensional Lorentzian Killing submersions. Our duality extends the Calabi correspondence between minimal graphs in the Euclidean space $\mathbb{R}^3$ and maximal graphs in the Lorentz-Minkowski spacetime $\mathbb{L}^3$, by allowing arbitrary prescribed mean curvature and bundle curvature. For instance, we transform the prescribed mean curvature equation in $\mathbb{L}^3$ into the minimal surface equation in the generalized Heisenberg space with prescribed bundle curvature. We present several applications of the twin correspondence to the study of the moduli space of complete spacelike surfaces in certain Lorentzian spacetimes.

math.DG↗
arXiv2012

The H-flow translating solitons in R^3 and R^4

Motivated by Ilmanen's correspondence, we present an explicit solution to the prescribed Hoffman-Osserman Gauss map problem for non-minimal translators to the mean curvature flow in Euclidean 4-space. We propose a conjecture on the non-existence of Jenkins-Serrin type unit-speed graphical translators.

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