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Jae Won Lee

Publications and source records attributed to Jae Won Lee.

4 recordsLinked to original sources
arXiv2026

Rigidity of Euclidean Minimal Hypersurfaces under Nonuniform Diagonal Dilations

Let $n\ge3$ and $D_t=\operatorname{diag}(t^{g_1},\ldots,t^{g_n})$ be a positive diagonal dilation family. We study connected embedded Euclidean hypersurfaces whose diagonal images are minimal. The level-set minimality operator splits into coefficients indexed by the pair sums $g_i+g_j$. Under pair-sum nonresonance, minimality at only $\binom n2$ distinct dilation parameters forces all pair coefficients to vanish. A dimension-reduction argument then shows, without any hypothesis on the coordinate components of the normal, that the second fundamental form vanishes identically. This yields an affine characterization. Repeated-weight helicoidal examples in every dimension and a resonant quadratic cone show that curvature cancellation can survive in genuinely nonuniform families. An application gives a finite-output-level rigidity criterion and an explicit representation for weighted-homogeneous production functions with minimal isoquants.

math.DG↗
arXiv2026

The Faraday Form of Conformal Products: Weyl Curvature and Four-Dimensional Einstein Rigidity

Let $(M^n,g)$ carry a conformal product structure whose adapted Weyl connection $D$ preserves orthogonal distributions of ranks $p,q\ge2$. We express the Faraday form $dθ$ directly in terms of the Weyl curvature of $g$. If $S$ is the orthogonal involution determined by the splitting and \[ \mathcal K_W(X)=\sum_i W_{X,e_i}(Se_i) \] for any local orthonormal frame, then \[ (dθ)^\sharp = \frac{n-2}{4(p-1)(q-1)}[\mathcal K_W,S]. \] No Ricci-curvature assumption is required. The same curvature calculation gives a companion formula for the symmetric part of $\nablaθ$, \[ SΘ^s-Θ^sS+θ^\sharp\wedge Sθ^\sharp = \frac{p-q}{n-2}(dθ)^\sharp +\frac1{n-2}[S,\Ric^\sharp], \qquad Θ=\nablaθ. \] When the Ricci tensor is block diagonal with respect to the product splitting, these formulas lead to explicit identities for the two components of the Lee form. In dimension four, the rank-$(2,2)$ splitting determines an ambi-Hermitian pair. The classical ambi-Hermitian curvature identities imply local closedness when the Ricci tensor is invariant under both complex structures; we also recover this conclusion from the mixed Weyl trace above. Finally, if $(M^4,g)$ is compact and Einstein, no specialness assumption is needed: $D=\nabla^g$. Hence the universal cover is the product of two simply connected surfaces with the same constant Gaussian curvature.

math.DG↗
arXiv2026

Partially Totally Real Submanifolds of Sasakian Manifolds

Partially totally real (PTR) submanifolds were introduced in Kähler geometry by distinguishing a totally real distribution and leaving its orthogonal complement unrestricted. In this paper we develop the corresponding framework for submanifolds of Sasakian manifolds tangent to the Reeb vector field. After separating the Reeb direction, we define the totally real and ambiguous distributions and show that anti-invariant, contact CR, hemi-slant and pointwise hemi-slant submanifolds occur as special cases of the Sasakian PTR framework. We establish the basic tangential and normal decompositions, study maximality and integrability, and derive the additional restrictions produced by the Reeb field. We then investigate the canonical morphisms \(P\) and \(F\), the geometry of the associated distributions, and PTR-submanifolds in Sasakian space forms. Explicit models are included to illustrate the principal structures and the differences from the Kähler case.

math.DG↗