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Quo-Shin Chi

Publications and source records attributed to Quo-Shin Chi.

At least 19 recordsLinked to original sources
arXiv2025

Classification of sextic curves in the Fano 3-fold $\mathcal{V}_5$ with rational Galois covers in ${\mathbb P}^3$

In this paper, we classify sextic curves in the Fano $3$-fold $\bf \mathcal{V}_5$ (the smooth quintic del Pezzo $3$-fold) that admit rational Galois covers in the complex ${\mathbb P}^3$. We show that the moduli space of such sextic curves is of complex dimension $2$ through the invariants of the engaged Galois groups for the explicit constructions. This raises the intriguing question of understanding the moduli space of sextic curves in ${\mathcal V}_5$ through their Galois covers in ${\mathbb P}^3$.

math.AG↗
arXiv2022

Fano 3-folds and classification of constantly curved holomorphic $2$-spheres of degree $6$ in the complex Grassmannian $G(2,5)$

Up to now the only known constantly curved sextic curve, i.e., holomorphic 2-sphere of degree 6, in the complex $G(2,5)$ has been the first associated curve of the Veronese curve of degree 4, which indicates that such curves are rare to find. Exploring the rich interplay between the ramification of harmonic sequences in differential geometry and algebro-geometric properties of projectively equivalent Fano 3-folds of index 2 and degree 5, we invoke the moduli space structure of sextic curves in the Fano 3-fold often referred to as $V_5$ to confirm the rarity of constancy of curvature, by establishing that the harmonic sequence of a generic sextic curve in $G(2, 5)$ is totally unramified. This paper proposes to investigate from the Galois viewpoint the way ramification can appear in relation to the constancy of curvature among nongeneric sextic curves in $G(2, 5)$. We prove through elaborate $PSL_2$-transvectant and engaged unitary analyses that, up to the ambient unitary equivalence, the moduli space of constantly curved sextic curves in $G(2,5)$ that are $GL(5,{\mathbb C})$-equivalent to those in $V_5$ ramified at the $PSL_2$-invariant 1-dimensional singular locus somewhere, is semialgebraic of dimension 2 all members of which barring the above Veronese curve are nonhomogeneous. Many explicit examples can be constructed.

math.DG↗
arXiv2020

The Isoparametric Story, a Heritage of Élie Cartan

In this article, we survey along the historical route the classification of isoparametric hypersurfaces in the sphere, paying attention to the employed techniques in the case of four principal curvatures.

math.DG↗
arXiv2020

Structure of minimal 2-spheres of constant curvature in the complex hyperquadric

In this paper, the singular-value decomposition theory of complex matrices is explored to study constantly curved 2-spheres minimal in both $\mathbb{C}P^n$ and the hyperquadric of $\mathbb{C}P^n$. The moduli space of all those noncongruent ones is introduced, which can be described by certain complex symmetric matrices modulo an appropriate group action. Using this description, many examples, such as constantly curved holomorphic 2-spheres of higher degree, nonhomogenous minimal 2-spheres of constant curvature, etc., are constructed. Uniqueness is proven for the totally real constantly curved 2-sphere minimal in both the hyperquadric and $\mathbb{C}P^n$.

math.DG↗
arXiv2017

Isoparametric hypersurfaces with four principal curvatures, IV

We prove that an isoparametric hypersurface with four principal curvatures and multiplicity pair $(7,8)$ is either the one constructed by Ozeki and Takeuchi, or one of the two constructed by Ferus, Karcher, and Münzner. This completes the classification of isoparametric hypersurfaces in spheres that É. Cartan initiated in the late 1930s.

math.DG↗
arXiv2014

Taut submanifolds are algebraic

We prove that every (compact) taut submanifold in Euclidean space is real algebraic, i.e., is a connected component of a real irreducible algebraic variety in the same ambient space. This answers affirmatively a question of Nicolaas Kuiper raised in the 1980s.

math.DG↗
arXiv2014

Ideal theory and classification of isoparametric hypersurfaces

The classification of isoparametric hypersurfaces with four principal curvatures in the sphere interplays in a deep fashion with commutative algebra, whose abstract and comprehensive nature might obscure a differential geometer's insight into the classification problem that encompasses a wide spectrum of geometry and topology. In this paper, we make an effort to bridge the gap by walking through the important part of commutative algebra central to the classification of such hypersurfaces, such that all the essential ideal-theoretic ingredients are laid out in a way as much intuitive, motivating and geometric with rigor maintained as possible. We then explain how we developed the technical side of the entailed ideal theory, pertinent to isoparametric hypersurfaces with four principal curvatures, for the classification done in our papers~\cite{CCJ},~\cite{Ch1} and~\cite{Ch3}.

math.DG↗
arXiv2011

Isoparametric hypersurfaces with four principal curvatures, III

The classification work [5], [9] left unsettled only those anomalous isoparametric hypersurfaces with four principal curvatures and multiplicity pair $\{4,5\},\{6,9\}$ or $\{7,8\}$ in the sphere. By systematically exploring the ideal theory in commutative algebra in conjunction with the geometry of isoparametric hypersurfaces, we show that an isoparametric hypersurface with four principal curvatures and multiplicities $\{4,5\}$ in $S^{19}$ is homogeneous, and, moreover, an isoparametric hypersurface with four principal curvatures and multiplicities $\{6,9\}$ in $S^{31}$ is either the inhomogeneous one constructed by Ferus, Karcher and Münzner, or the one that is homogeneous. This classification reveals the striking resemblance between these two rather different types of isoparametric hypersurfaces in the homogeneous category, even though the one with multiplicities $\{6,9\}$ is of the type constructed by Ferus, Karcher and Münzner and the one with multiplicities $\{4,5\}$ stands alone by itself. The quaternion and the octonion algebras play a fundamental role in their geometric structures. A unifying theme in [5]. [9] and the present sequel to them is Serre's criterion of normal varieties. Its technical side pertinent to our situation that we developed in [5], [9] and extend in this sequel is instrumental. The classification leaves only the case of multiplicity pair $\{7,8\}$ open.

math.DG↗
arXiv2010

Isoparametric hypersurfaces with four principal curvatures, II

In this sequel, employing more commutative algebra than that explored in \cite{CCJ}, we show that an isoparametric hypersurface with four principal curvatures and multiplicities $(3,4)$ in $S^{15}$ is one constructed by Ozeki-Takeuchi \cite[I]{OT} and Ferus-Karcher-Münzner \cite{FKM}, referred to collectively as of OT-FKM type. In fact, this new approach also gives a considerably simpler, both structurally and technically, proof \cite{CCJ} that an isoparametric hypersurface with four principal curvatures in spheres with the multiplicity constraint $m_2\geq 2m_1-1$ is of OT-FKM type, which left unsettled exactly the four anomalous multiplicity pairs $(4,5),(3,4),(7,8)$ and $(6,9)$, where the last three are closely tied, respectively, with the quaternion algebra, the octonion algebra and the complexified octonion algebra, whereas the first stands alone by itself in that it cannot be of OT-FKM type. A byproduct of this new approach is that we see that Condition B, introduced by Ozeki and Takeuchi \cite[I]{OT} in their construction of inhomogeneous isoparametric hypersurfaces, naturally arises. The cases for the multiplicity pairs $(4,5),(6,9)$ and $(7,8)$ remain open now.

math.DG↗
arXiv2009

A new look at Condition A

Ozeki and Takeuchi \cite[I]{OT} introduced the notion of Condition A and Condition B to construct two classes of inhomogeneous isoparametric hypersurfaces with four principal curvatures in spheres, which were later generalized by Ferus, Karcher and Münzner to many more examples via the Clifford representations; we will refer to these examples of Ozeki and Takeuchi and of Ferus, Karcher and Münzner collectively as OT-FKM type throughout the paper. Dorfmeister and Neher \cite{DN} then employed isoparametric triple systems \cite{DN1}, which are algebraic in nature, to prove that Condition A alone implies the isoparametric hypersurface is of OT-FKM type. Their proof for the case of multiplicity pairs $\{3,4\}$ and $\{7,8\}$ rests on a fairly involved algebraic classification result \cite{Mc} about composition triples. In light of the classification \cite{CCJ} that leaves only the four exceptional multiplicity pairs $\{4,5\},\{3,4\},\{7,8\}$ and $\{6,9\}$ unsettled, it appears that Condition A may hold the key to the classification when the multiplicity pairs are $\{3,4\}$ and $\{7,8\}$. Thus Condition A deserves to be scrutinized and understood more thoroughly from different angles. In this paper, we give a fairly short and rather straightforward proof of the result of Dorfmeister and Neher, with emphasis on the multiplicity pairs $\{3,4\}$ and $\{7,8\}$, based on more geometric considerations. We make it explicit and apparent that the octonian algebra governs the underlying isoparametric structure.

math.DG↗
arXiv2008

Isoparametric hypersurfaces with four principal curvatures revisited

The classification of isoparametric hypersurfaces with four principal curvatures in spheres in [2] hinges on a crucial characterization, in terms of four sets of equations of the 2nd fundamental form tensors of a focal submanifold, of an isoparametric hypersurface of the type constructed by Ferus, Karcher and Münzner. The proof of the characterization in [2] is an extremely long calculation by exterior derivatives with remarkable cancellations, which is motivated by the idea that an isoparametric hypersurface is defined by an over-determined system of partial differential equations. Therefore, exterior differentiating sufficiently many times should gather us enough information for the conclusion. In spite of its elementary nature, the magnitude of the calculation and the surprisingly pleasant cancellations make it desirable to understand the underlying geometric principles. In this paper, we give a conceptual, and considerably shorter, proof of the characterization based on Ozeki and Takeuchi's expansion formula for the Cartan-Münzner polynomial. Along the way the geometric meaning of these four sets of equations also becomes clear.

math.DG↗
arXiv2007

On Kuiper's conjecture

We prove that any connected proper Dupin hypersurface in $\R^n$ is analytic algebraic and is an open subset of a connected component of an irreducible algebraic set. We prove the same result for any connected non-proper Dupin hypersurface in $\R^n$ that satisfies a certain finiteness condition. Hence any taut submanifold M in $\R^n$, whose tube $M_ε$ satisfies this finiteness condition, is analytic algebraic and is a connected component of an irreducible algebraic set. In particular, we prove that every taut submanifold of dimension $m \leq 4$ is algebraic.

math.DG↗
arXiv2005

Dupin hypersurfaces with four principal curvatures, II

If $M$ is an isoparametric hypersurface in a sphere $S^n$ with four distrinct principal curvatures, then the principal curvatures $κ_1,...,κ_4$ can be ordered so that their multiplicities satisfy $m_1=m_2$ and $m_3=m_4$, and the cross-ratio $r$ of the principal curvatures (the Lie curvature) equals -1. In this paper, we prove that if $M$ is an irreducible connected proper Dupin hypersurface in $\R^n$ (or $S^n$) with four distinct principal curvatures with multiplicities $m_1=m_2 \geq 1$ and $m_3=m_4=1$, and constant Lie curvature $r=-1$, then $M$ is equivalent by Lie sphere transformation to an isoparametric hypersurface in a sphere. This result remains true if the assumption of irreducibility is replaced by compactness and $r$ is merely assumed to be constant.

math.DG↗
arXiv2005

Strong Analytic Controllability for Hydrogen Control Systems

The realization and representation of so(4,2) associated with the hydrogen atom Hamiltonian are derived. By choosing operators from the realization of so(4,2) as interacting Hamiltonians, a hydrogen atom control system is constructed, and it is proved that this control system is strongly analytically controllable based on a time-dependent strong analytic controllability theorem.

quant-ph↗
arXiv2004

Analytic Controllability of Time-Dependent Quantum Control Systems

The question of controllability is investigated for a quantum control system in which the Hamiltonian operator components carry explicit time dependence which is not under the control of an external agent. We consider the general situation in which the state moves in an infinite-dimensional Hilbert space, a drift term is present, and the operators driving the state evolution may be unbounded. However, considerations are restricted by the assumption that there exists an analytic domain, dense in the state space, on which solutions of the controlled Schrodinger equation may be expressed globally in exponential form. The issue of controllability then naturally focuses on the ability to steer the quantum state on a finite-dimensional submanifold of the unit sphere in Hilbert space -- and thus on analytic controllability. A relatively straightforward strategy allows the extension of Lie-algebraic conditions for strong analytic controllability derived earlier for the simpler, time-independent system in which the drift Hamiltonian and the interaction Hamiltonia have no intrinsic time dependence. Enlarging the state space by one dimension corresponding to the time variable, we construct an augmented control system that can be treated as time-independent. Methods developed by Kunita can then be implemented to establish controllability conditions for the one-dimension-reduced system defined by the original time-dependent Schrodinger control problem. The applicability of the resulting theorem is illustrated with selected examples.

quant-ph↗
arXiv2004

Isoparametric hypersurfaces with four principal curvatures

Let $M$ be an isoparametric hypersurface in the sphere $S^n$ with four distinct principal curvatures. Münzner showed that the four principal curvatures can have at most two distinct multiplicities $m_1, m_2$, and Stolz showed that the pair $(m_1,m_2)$ must either be $(2,2)$, $(4,5)$, or be equal to the multiplicities of an isoparametric hypersurface of FKM-type, constructed by Ferus, Karcher and Münzner from orthogonal representations of Clifford algebras. In this paper, we prove that if the multiplicities satisfy $m_2 \geq 3m_1 - 1$, then the isoparametric hypersurface $M$ must be of FKM-type. Together with known results of Takagi for the case $m_1 = 1$, and Ozeki and Takeuchi for $m_1 = 2$, this handles all possible pairs of multiplicities except for 10 cases, for which the classification problem remains open. The paper improves the result of a pre-existing preprint with the same title, in which 14 cases remained open.

math.DG↗
arXiv1995

On the Incompleteness of Berger's List of Holonomy Representations

In 1955, Berger \cite{Ber} gave a list of irreducible reductive representations which can occur as the holonomy of a torsion-free affine connection. This list was stated to be complete up to possibly a finite number of missing entries. In this paper, we show that there is, in fact, an infinite family of representations which are missing from this list, thereby showing the incompleteness of Berger's classification. Moreover, we develop a method to construct torsion-free connections with prescribed holonomy, and use it to give a complete description of the torsion-free affine connections with these new holonomies. We also deduce some striking facts about their global behaviour.

dg-ga↗