Homogeneous and inhomogeneous isoparametric hypersurfaces in - PowerPoint PPT Presentation

Isoparametric hypersurfaces in space forms Theorem [Cartan (1939), Segre (1938)] Let M be a hypersurface in a real space form ¯ M ∈ { R n , R H n , S n } . Then: M is isoparametric ⇔ M has constant principal curvatures If ¯ M ∈ { R n , R H n } , M is isoparametric ⇔ M is homogeneous Classification in the real hyperbolic space R H n [Cartan (1939)] Tot. geod. R H n − 1 and equidistant Tubes around a Geodesic spheres Horospheres hypersurfaces tot. geod. R H k

Isoparametric hypersurfaces in space forms Theorem [Cartan (1939), Segre (1938)] Let M be a hypersurface in a real space form ¯ M ∈ { R n , R H n , S n } . Then: M is isoparametric ⇔ M has constant principal curvatures If ¯ M ∈ { R n , R H n } , M is isoparametric ⇔ M is homogeneous Classification in spheres S n There are inhomogeneous examples [Ferus, Karcher, M¨ unzner (1981)] All isoparametric hypersurfaces are homogeneous or of FKM-type [Cartan; M¨ unzner; Takagi; Ozeki, Takeuchi; Tang; Fang; Stolz; Cecil, Chi, Jensen; Immervoll; Abresch; Dorfmeister, Neher; Miyaoka; Chi]

Isoparametric hypersurfaces in nonconstant curvature General geometric and topological structure results [Wang (1987), Ge, Tang (2013), Ge, Tang, Yan (2015), Qian, Tang (2015), Ge, Radeschi (2015), Ge (2016)]

Isoparametric hypersurfaces in nonconstant curvature General geometric and topological structure results [Wang (1987), Ge, Tang (2013), Ge, Tang, Yan (2015), Qian, Tang (2015), Ge, Radeschi (2015), Ge (2016)] Classification in C P n , n � = 15 [DV (2016)] and in H P n , n � = 7 [DV, Gorodski (2018)] There are countably many inhomogeneous examples, all of them with nonconstant principal curvatures

Isoparametric hypersurfaces in nonconstant curvature General geometric and topological structure results [Wang (1987), Ge, Tang (2013), Ge, Tang, Yan (2015), Qian, Tang (2015), Ge, Radeschi (2015), Ge (2016)] Classification in C P n , n � = 15 [DV (2016)] and in H P n , n � = 7 [DV, Gorodski (2018)] There are countably many inhomogeneous examples, all of them with nonconstant principal curvatures Classification in S 2 × S 2 [Urbano (2016)] and in E ( κ, τ )-spaces [DV, Manzano (2018)] In these two cases, all examples are homogeneous

Isoparametric hypersurfaces in nonconstant curvature General geometric and topological structure results [Wang (1987), Ge, Tang (2013), Ge, Tang, Yan (2015), Qian, Tang (2015), Ge, Radeschi (2015), Ge (2016)] Classification in C P n , n � = 15 [DV (2016)] and in H P n , n � = 7 [DV, Gorodski (2018)] There are countably many inhomogeneous examples, all of them with nonconstant principal curvatures Classification in S 2 × S 2 [Urbano (2016)] and in E ( κ, τ )-spaces [DV, Manzano (2018)] In these two cases, all examples are homogeneous Question What happens in symmetric spaces of noncompact type?

Symmetric spaces of noncompact type 1 Homogeneous and isoparametric hypersurfaces 2 Symmetric spaces of noncompact type and rank one Cohomogeneity one actions 1 Isoparametric hypersurfaces 2 3 The quaternionic hyperbolic space

Symmetric spaces of noncompact type Definition [Cartan (1926)] A symmetric space is a Riemannian manifold ¯ M whose geodesic symmetry σ p : exp p ( v ) �→ exp p ( − v ), v ∈ T p ¯ M , around each p ∈ ¯ M is a global isometry of ¯ M .

Symmetric spaces of noncompact type Definition [Cartan (1926)] A symmetric space is a Riemannian manifold ¯ M whose geodesic symmetry σ p : exp p ( v ) �→ exp p ( − v ), v ∈ T p ¯ M , around each p ∈ ¯ M is a global isometry of ¯ M .

Symmetric spaces of noncompact type Definition [Cartan (1926)] A symmetric space is a Riemannian manifold ¯ M whose geodesic symmetry σ p : exp p ( v ) �→ exp p ( − v ), v ∈ T p ¯ M , around each p ∈ ¯ M is a global isometry of ¯ M .

Symmetric spaces of noncompact type Definition [Cartan (1926)] A symmetric space is a Riemannian manifold ¯ M whose geodesic symmetry σ p : exp p ( v ) �→ exp p ( − v ), v ∈ T p ¯ M , around each p ∈ ¯ M is a global isometry of ¯ M .

Symmetric spaces of noncompact type Definition [Cartan (1926)] A symmetric space is a Riemannian manifold ¯ M whose geodesic symmetry σ p : exp p ( v ) �→ exp p ( − v ), v ∈ T p ¯ M , around each p ∈ ¯ M is a global isometry of ¯ M .

Symmetric spaces of noncompact type Definition [Cartan (1926)] A symmetric space is a Riemannian manifold ¯ M whose geodesic symmetry σ p : exp p ( v ) �→ exp p ( − v ), v ∈ T p ¯ M , around each p ∈ ¯ M is a global isometry of ¯ M .

Symmetric spaces of noncompact type Definition [Cartan (1926)] A symmetric space is a Riemannian manifold ¯ M whose geodesic symmetry σ p : exp p ( v ) �→ exp p ( − v ), v ∈ T p ¯ M , around each p ∈ ¯ M is a global isometry of ¯ M .

Symmetric spaces of noncompact type Definition [Cartan (1926)] A symmetric space is a Riemannian manifold ¯ M whose geodesic symmetry σ p : exp p ( v ) �→ exp p ( − v ), v ∈ T p ¯ M , around each p ∈ ¯ M is a global isometry of ¯ M . Symmetric spaces are complete and homogeneous

Symmetric spaces of noncompact type Definition [Cartan (1926)] A symmetric space is a Riemannian manifold ¯ M whose geodesic symmetry σ p : exp p ( v ) �→ exp p ( − v ), v ∈ T p ¯ M , around each p ∈ ¯ M is a global isometry of ¯ M . Symmetric spaces are complete and homogeneous M ∼ ¯ = G / K , where G = Isom 0 ( ¯ M ) and K = { g ∈ G : g ( o ) = o } are Lie groups, and o ∈ ¯ M is a base point

Symmetric spaces of noncompact type Definition [Cartan (1926)] A symmetric space is a Riemannian manifold ¯ M whose geodesic symmetry σ p : exp p ( v ) �→ exp p ( − v ), v ∈ T p ¯ M , around each p ∈ ¯ M is a global isometry of ¯ M . Symmetric spaces are complete and homogeneous M ∼ ¯ = G / K , where G = Isom 0 ( ¯ M ) and K = { g ∈ G : g ( o ) = o } are Lie groups, and o ∈ ¯ M is a base point compact type noncompact type Euclidean type duality

Symmetric spaces of noncompact type Definition [Cartan (1926)] A symmetric space is a Riemannian manifold ¯ M whose geodesic symmetry σ p : exp p ( v ) �→ exp p ( − v ), v ∈ T p ¯ M , around each p ∈ ¯ M is a global isometry of ¯ M . Symmetric spaces are complete and homogeneous M ∼ ¯ = G / K , where G = Isom 0 ( ¯ M ) and K = { g ∈ G : g ( o ) = o } are Lie groups, and o ∈ ¯ M is a base point compact type noncompact type Euclidean type duality ¯ ¯ M compact, M noncompact, ¯ M = R n / Γ flat sec( ¯ sec( ¯ M ) ≥ 0, M ) ≤ 0, g compact semisimple g noncompact semisimple

Symmetric spaces of noncompact type M ∼ ¯ = G / K symmetric space of noncompact type

Symmetric spaces of noncompact type diffeo. M ∼ ∼ ¯ ⇒ ¯ = B n = G / K symmetric space of noncompact type = M

Symmetric spaces of noncompact type diffeo. M ∼ ∼ ¯ ⇒ ¯ = B n = G / K symmetric space of noncompact type = M p ∼ = T o ¯ g = k ⊕ p Cartan decomposition, M

Symmetric spaces of noncompact type diffeo. M ∼ ∼ ¯ ⇒ ¯ = B n = G / K symmetric space of noncompact type = M p ∼ = T o ¯ g = k ⊕ p Cartan decomposition, M rank ¯ a maximal abelian subspace of p , M := dim a

Symmetric spaces of noncompact type diffeo. M ∼ ∼ ¯ ⇒ ¯ = B n = G / K symmetric space of noncompact type = M p ∼ = T o ¯ g = k ⊕ p Cartan decomposition, M rank ¯ a maximal abelian subspace of p , M := dim a Iwasawa decomposition g = k ⊕ a ⊕ n n is nilpotent

Symmetric spaces of noncompact type diffeo. M ∼ ∼ ¯ ⇒ ¯ = B n = G / K symmetric space of noncompact type = M p ∼ = T o ¯ g = k ⊕ p Cartan decomposition, M rank ¯ a maximal abelian subspace of p , M := dim a Iwasawa decomposition g = k ⊕ a ⊕ n n is nilpotent diffeo. ∼ G = K × A × N N K A

Symmetric spaces of noncompact type diffeo. M ∼ ∼ ¯ ⇒ ¯ = B n = G / K symmetric space of noncompact type = M p ∼ = T o ¯ g = k ⊕ p Cartan decomposition, M rank ¯ a maximal abelian subspace of p , M := dim a Iwasawa decomposition g = k ⊕ a ⊕ n n is nilpotent diffeo. ∼ G = K × A × N N K A a ⊕ n Lie subalgebra of g ❀ AN Lie subgroup of G

Symmetric spaces of noncompact type diffeo. M ∼ ∼ ¯ ⇒ ¯ = B n = G / K symmetric space of noncompact type = M p ∼ = T o ¯ g = k ⊕ p Cartan decomposition, M rank ¯ a maximal abelian subspace of p , M := dim a Iwasawa decomposition g = k ⊕ a ⊕ n n is nilpotent diffeo. ∼ G = K × A × N N K A a ⊕ n Lie subalgebra of g ❀ AN Lie subgroup of G AN acts freely and transitively on ¯ M ❀ AN is diffeomorphic to ¯ M

Symmetric spaces of noncompact type diffeo. M ∼ ∼ ¯ ⇒ ¯ = B n = G / K symmetric space of noncompact type = M p ∼ = T o ¯ g = k ⊕ p Cartan decomposition, M rank ¯ a maximal abelian subspace of p , M := dim a Iwasawa decomposition g = k ⊕ a ⊕ n n is nilpotent diffeo. ∼ G = K × A × N N K A a ⊕ n Lie subalgebra of g ❀ AN Lie subgroup of G AN acts freely and transitively on ¯ M ❀ AN is diffeomorphic to ¯ M The solvable model of a symmetric space of noncompact type ¯ M is isometric to AN endowed with a left-invariant metric.

Symmetric spaces of noncompact type and rank one isom. M ∼ ¯ ∼ = AN symmetric space of noncompact type, rank ¯ = G / K M = 1

Symmetric spaces of noncompact type and rank one isom. M ∼ ¯ ∼ = AN symmetric space of noncompact type, rank ¯ = G / K M = 1 a ∼ a ⊕ n = a ⊕ v ⊕ z , = R , z = Z ( n )

Symmetric spaces of noncompact type and rank one isom. M ∼ ¯ ∼ = AN symmetric space of noncompact type, rank ¯ = G / K M = 1 a ∼ a ⊕ n = a ⊕ v ⊕ z , = R , z = Z ( n ) Symmetric spaces of noncompact type and rank 1 R H n C H n H H n O H 2 ¯ M F − 20 SO 0 (1 , n ) SU(1 , n ) Sp(1 , n ) 4 SO( n ) S(U(1) × U( n )) Sp(1) × Sp( n ) Spin(9) R n − 1 C n − 1 H n − 1 O v dim z 0 1 3 7

Symmetric spaces of noncompact type 1 Homogeneous and isoparametric hypersurfaces 2 Symmetric spaces of noncompact type and rank one Cohomogeneity one actions 1 Isoparametric hypersurfaces 2 3 The quaternionic hyperbolic space

Cohomogeneity one actions on hyperbolic spaces F H n symmetric space of noncompact type and rank one, F ∈ { R , C , H , O } Cohomogeneity one actions with a totally geodesic singular orbit [Berndt, Br¨ uck (2001)] Tubes around totally geodesic submanifolds P in F H n are homogeneous if and only if in R H n : P = { point } , R H 1 , . . . , R H n − 1 in C H n : P = { point } , C H 1 , . . . , C H n − 1 , R H n in H H n : P = { point } , H H 1 , . . . , H H n − 1 , C H n P in O H 2 : P = { point } , O H 1 , H H 2

Cohomogeneity one actions on hyperbolic spaces isom. F H n ∼ ∼ = G / K = AN symmetric space of noncompact type and rank one a ∼ a ⊕ n = a ⊕ v ⊕ z , = R , z = Z ( n ), K 0 = N K ( a ) Symmetric spaces of noncompact type and rank 1 O H 2 F H n R H n C H n H H n R n − 1 C n − 1 H n − 1 v O

Cohomogeneity one actions on hyperbolic spaces isom. F H n ∼ ∼ = G / K = AN symmetric space of noncompact type and rank one a ∼ a ⊕ n = a ⊕ v ⊕ z , = R , z = Z ( n ), K 0 = N K ( a ) Symmetric spaces of noncompact type and rank 1 F H n R H n C H n H H n O H 2 R n − 1 C n − 1 H n − 1 v O Cohomogeneity one actions without singular orbits [Berndt, Br¨ uck (2001), Berndt, Tamaru (2003)] Orbit equivalent to the action of: N ❀ horosphere foliation The connected subgroup of G with Lie algebra a ⊕ w ⊕ z , where w is a (real) hyperplane in v

Cohomogeneity one actions on hyperbolic spaces isom. F H n ∼ ∼ = G / K = AN symmetric space of noncompact type and rank one a ∼ a ⊕ n = a ⊕ v ⊕ z , = R , z = Z ( n ), K 0 = N K ( a ) Symmetric spaces of noncompact type and rank 1 F H n R H n C H n H H n O H 2 R n − 1 C n − 1 H n − 1 O v Cohomogeneity one actions with a non-totally singular orbit [Berndt, Br¨ uck (2001)] w � v (real) subspace = ⇒ s w = a ⊕ w ⊕ z is a Lie algebra S w connected subgroup of AN with Lie algebra s w

Cohomogeneity one actions on hyperbolic spaces isom. F H n ∼ ∼ = G / K = AN symmetric space of noncompact type and rank one a ∼ a ⊕ n = a ⊕ v ⊕ z , = R , z = Z ( n ), K 0 = N K ( a ) Symmetric spaces of noncompact type and rank 1 F H n R H n C H n H H n O H 2 R n − 1 C n − 1 H n − 1 O v Cohomogeneity one actions with a non-totally singular orbit [Berndt, Br¨ uck (2001)] w � v (real) subspace = ⇒ s w = a ⊕ w ⊕ z is a Lie algebra S w S w connected subgroup of AN with Lie algebra s w The tubes around S w are homogeneous if and only if tube N K 0 ( w ) acts transitively on the unit sphere of w ⊥ (the orthogonal complement of w in v )

Cohomogeneity one actions on hyperbolic spaces isom. F H n ∼ ∼ = G / K = AN symmetric space of noncompact type and rank one a ∼ a ⊕ n = a ⊕ v ⊕ z , = R , z = Z ( n ), K 0 = N K ( a ) Symmetric spaces of noncompact type and rank 1 F H n R H n C H n H H n O H 2 R n − 1 C n − 1 H n − 1 O v Cohomogeneity one actions with a non-totally singular orbit [Berndt, Br¨ uck (2001)] w � v (real) subspace = ⇒ s w = a ⊕ w ⊕ z is a Lie algebra S w connected subgroup of AN with Lie algebra s w S w The tubes around S w are homogeneous if and only if N K 0 ( w ) acts transitively on the unit sphere of w ⊥ (the orthogonal complement of w in v )

Cohomogeneity one actions on hyperbolic spaces isom. ∼ F H n = AN symmetric space of noncompact type and rank one a ∼ a ⊕ n = a ⊕ v ⊕ z , = R , z = Z ( n ), K 0 = N K ( a )

Cohomogeneity one actions on hyperbolic spaces isom. ∼ F H n = AN symmetric space of noncompact type and rank one a ∼ a ⊕ n = a ⊕ v ⊕ z , = R , z = Z ( n ), K 0 = N K ( a ) Theorem [Berndt, Tamaru (2007)] For a cohomogeneity one action on F H n , one of the following holds: There is a totally geodesic singular orbit.

Cohomogeneity one actions on hyperbolic spaces isom. ∼ F H n = AN symmetric space of noncompact type and rank one a ∼ a ⊕ n = a ⊕ v ⊕ z , = R , z = Z ( n ), K 0 = N K ( a ) Theorem [Berndt, Tamaru (2007)] For a cohomogeneity one action on F H n , one of the following holds: There is a totally geodesic singular orbit. �

Cohomogeneity one actions on hyperbolic spaces isom. ∼ F H n = AN symmetric space of noncompact type and rank one a ∼ a ⊕ n = a ⊕ v ⊕ z , = R , z = Z ( n ), K 0 = N K ( a ) Theorem [Berndt, Tamaru (2007)] For a cohomogeneity one action on F H n , one of the following holds: There is a totally geodesic singular orbit. � Its orbit foliation is regular.

Cohomogeneity one actions on hyperbolic spaces isom. ∼ F H n = AN symmetric space of noncompact type and rank one a ∼ a ⊕ n = a ⊕ v ⊕ z , = R , z = Z ( n ), K 0 = N K ( a ) Theorem [Berndt, Tamaru (2007)] For a cohomogeneity one action on F H n , one of the following holds: There is a totally geodesic singular orbit. � Its orbit foliation is regular. �

Cohomogeneity one actions on hyperbolic spaces isom. ∼ F H n = AN symmetric space of noncompact type and rank one a ∼ a ⊕ n = a ⊕ v ⊕ z , = R , z = Z ( n ), K 0 = N K ( a ) Theorem [Berndt, Tamaru (2007)] For a cohomogeneity one action on F H n , one of the following holds: There is a totally geodesic singular orbit. � Its orbit foliation is regular. � There is a non-totally geodesic singular orbit S w , where w � v is such that N K 0 ( w ) acts transitively on the unit sphere of w ⊥ .

Cohomogeneity one actions on hyperbolic spaces isom. ∼ F H n = AN symmetric space of noncompact type and rank one a ∼ a ⊕ n = a ⊕ v ⊕ z , = R , z = Z ( n ), K 0 = N K ( a ) Theorem [Berndt, Tamaru (2007)] For a cohomogeneity one action on F H n , one of the following holds: There is a totally geodesic singular orbit. � Its orbit foliation is regular. � There is a non-totally geodesic singular orbit S w , where w � v is such that N K 0 ( w ) acts transitively on the unit sphere of w ⊥ . The study of the last case was carried out for R H n , C H n , H H 2 and O H 2

Cohomogeneity one actions on hyperbolic spaces isom. ∼ F H n = AN symmetric space of noncompact type and rank one a ∼ a ⊕ n = a ⊕ v ⊕ z , = R , z = Z ( n ), K 0 = N K ( a ) Theorem [Berndt, Tamaru (2007)] For a cohomogeneity one action on F H n , one of the following holds: There is a totally geodesic singular orbit. � Its orbit foliation is regular. � There is a non-totally geodesic singular orbit S w , where w � v is such that N K 0 ( w ) acts transitively on the unit sphere of w ⊥ . The study of the last case was carried out for R H n , C H n , H H 2 and O H 2 Problem Analyze the last case for H H n , n ≥ 3, to conclude the classification.

Symmetric spaces of noncompact type 1 Homogeneous and isoparametric hypersurfaces 2 Symmetric spaces of noncompact type and rank one Cohomogeneity one actions 1 Isoparametric hypersurfaces 2 3 The quaternionic hyperbolic space

New isoparametric hypersurfaces isom. ∼ F H n = AN symmetric space of noncompact type and rank one a ∼ a ⊕ n = a ⊕ v ⊕ z , = R , z = Z ( n )

New isoparametric hypersurfaces isom. ∼ F H n = AN symmetric space of noncompact type and rank one a ∼ a ⊕ n = a ⊕ v ⊕ z , = R , z = Z ( n ) New isoparametric hypersurfaces [D´ ıaz-Ramos, DV (2013)] w � v real subspace = ⇒ s w = a ⊕ w ⊕ z is a Lie algebra

New isoparametric hypersurfaces isom. ∼ F H n = AN symmetric space of noncompact type and rank one a ∼ a ⊕ n = a ⊕ v ⊕ z , = R , z = Z ( n ) New isoparametric hypersurfaces [D´ ıaz-Ramos, DV (2013)] w � v real subspace = ⇒ s w = a ⊕ w ⊕ z is a Lie algebra S w connected subgroup of AN with Lie algebra s w

New isoparametric hypersurfaces isom. ∼ F H n = AN symmetric space of noncompact type and rank one a ∼ a ⊕ n = a ⊕ v ⊕ z , = R , z = Z ( n ) New isoparametric hypersurfaces [D´ ıaz-Ramos, DV (2013)] w � v real subspace = ⇒ s w = a ⊕ w ⊕ z is a Lie algebra S w connected subgroup of AN with Lie algebra s w S w is a homogeneous minimal submanifold

New isoparametric hypersurfaces isom. ∼ F H n = AN symmetric space of noncompact type and rank one a ∼ a ⊕ n = a ⊕ v ⊕ z , = R , z = Z ( n ) New isoparametric hypersurfaces [D´ ıaz-Ramos, DV (2013)] w � v real subspace = ⇒ s w = a ⊕ w ⊕ z is a Lie S w algebra S w connected subgroup of AN with Lie algebra s w S w is a homogeneous minimal submanifold tube The tubes around S w are isoparametric

New isoparametric hypersurfaces isom. ∼ F H n = AN symmetric space of noncompact type and rank one a ∼ a ⊕ n = a ⊕ v ⊕ z , = R , z = Z ( n ) New isoparametric hypersurfaces [D´ ıaz-Ramos, DV (2013)] w � v real subspace = ⇒ s w = a ⊕ w ⊕ z is a Lie algebra S w connected subgroup of AN with Lie algebra s w S w S w is a homogeneous minimal submanifold The tubes around S w are isoparametric In R H n such hypersurfaces are homogeneous

New isoparametric hypersurfaces isom. ∼ F H n = AN symmetric space of noncompact type and rank one a ∼ a ⊕ n = a ⊕ v ⊕ z , = R , z = Z ( n ) New isoparametric hypersurfaces [D´ ıaz-Ramos, DV (2013)] w � v real subspace = ⇒ s w = a ⊕ w ⊕ z is a Lie algebra S w connected subgroup of AN with Lie algebra s w S w S w is a homogeneous minimal submanifold The tubes around S w are isoparametric In R H n such hypersurfaces are homogeneous In C H n and H H n , n ≥ 3, there are inhomogeneous isoparametric families of hypersurfaces with nonconstant principal curvatures

New isoparametric hypersurfaces isom. ∼ F H n = AN symmetric space of noncompact type and rank one a ∼ a ⊕ n = a ⊕ v ⊕ z , = R , z = Z ( n ) New isoparametric hypersurfaces [D´ ıaz-Ramos, DV (2013)] w � v real subspace = ⇒ s w = a ⊕ w ⊕ z is a Lie algebra S w connected subgroup of AN with Lie algebra s w S w S w is a homogeneous minimal submanifold The tubes around S w are isoparametric In R H n such hypersurfaces are homogeneous In C H n and H H n , n ≥ 3, there are inhomogeneous isoparametric families of hypersurfaces with nonconstant principal curvatures In O H 2 there is one inhomogeneous isoparametric family of hypersurfaces with constant principal curvatures (when dim w = 3)

Classification in the complex hyperbolic space Theorem [D´ ıaz-Ramos, DV, Sanmart´ ın-L´ opez (2017)] A connected hypersurface M in the complex hyperbolic space C H n is isoparametric if and only if it is an open subset of: A tube around a totally geodesic complex hyperbolic space C H k A tube around a totally geodesic real hyperbolic space R H n A horosphere A tube around a homogeneous minimal submanifold S w

Classification in the complex hyperbolic space Theorem [D´ ıaz-Ramos, DV, Sanmart´ ın-L´ opez (2017)] A connected hypersurface M in the complex hyperbolic space C H n is isoparametric if and only if it is an open subset of: A tube around a totally geodesic complex hyperbolic space C H k A tube around a totally geodesic real hyperbolic space R H n A horosphere A tube around a homogeneous minimal submanifold S w Classical examples [Montiel (1985)]: all are homogeneous

Classification in the complex hyperbolic space Theorem [D´ ıaz-Ramos, DV, Sanmart´ ın-L´ opez (2017)] A connected hypersurface M in the complex hyperbolic space C H n is isoparametric if and only if it is an open subset of: A tube around a totally geodesic complex hyperbolic space C H k A tube around a totally geodesic real hyperbolic space R H n A horosphere A tube around a homogeneous minimal submanifold S w Classical examples [Montiel (1985)]: all are homogeneous New examples: there are both (uncountably many) homogeneous [Berndt, Br¨ uck (2001)] and inhomogeneous [D´ ıaz-Ramos, DV (2012)] examples, depending on w ⊂ v

The quaternionic hyperbolic space 1 Homogeneous and isoparametric hypersurfaces 2 Symmetric spaces of noncompact type and rank one Cohomogeneity one actions 1 Isoparametric hypersurfaces 2 3 The quaternionic hyperbolic space

The quaternionic hyperbolic space Problem Classify cohomogeneity one actions on H H n +1 , n ≥ 2.

The quaternionic hyperbolic space Problem Classify cohomogeneity one actions on H H n +1 , n ≥ 2. Equivalent problem [Berndt, Tamaru (2007)] = H n such that N K 0 ( w ) acts transitively on Classify real subspaces w ⊂ v ∼ the unit sphere of w ⊥ .

The quaternionic hyperbolic space Problem Classify cohomogeneity one actions on H H n +1 , n ≥ 2. Equivalent problem [Berndt, Tamaru (2007)] = H n such that N K 0 ( w ) acts transitively on Classify real subspaces w ⊂ v ∼ the unit sphere of w ⊥ . = H n via ( A , q ) · v = Avq − 1 K 0 ∼ = Sp( n )Sp(1) acts on v ∼

The quaternionic hyperbolic space Problem Classify cohomogeneity one actions on H H n +1 , n ≥ 2. Equivalent problem [Berndt, Tamaru (2007)] = H n such that N K 0 ( w ) acts transitively on Classify real subspaces w ⊂ v ∼ the unit sphere of w ⊥ . = H n via ( A , q ) · v = Avq − 1 K 0 ∼ = Sp( n )Sp(1) acts on v ∼ Definition A real subspace V of H n is protohomogeneous if there is a (connected) subgroup of Sp( n )Sp(1) that acts transitively on the unit sphere of V .

The quaternionic hyperbolic space Problem Classify cohomogeneity one actions on H H n +1 , n ≥ 2. Equivalent problem [Berndt, Tamaru (2007)] = H n such that N K 0 ( w ) acts transitively on Classify real subspaces w ⊂ v ∼ the unit sphere of w ⊥ . = H n via ( A , q ) · v = Avq − 1 K 0 ∼ = Sp( n )Sp(1) acts on v ∼ Definition A real subspace V of H n is protohomogeneous if there is a (connected) subgroup of Sp( n )Sp(1) that acts transitively on the unit sphere of V . Equivalent problem Classify protohomogeneous subspaces of H n .

Getting intuition in the complex setting Analogous definition in C n A real subspace V of C n is protohomogeneous if there is a (connected) subgroup of U( n ) that acts transitively on the unit sphere of V Analogous problem in C n Classify protohomogeneous subspaces of C n

Getting intuition in the complex setting Analogous definition in C n A real subspace V of C n is protohomogeneous if there is a (connected) subgroup of U( n ) that acts transitively on the unit sphere of V Analogous problem in C n Classify protohomogeneous subspaces of C n { e 1 , . . . , e n } C -orthonormal basis of C n , J complex structure of C n

Getting intuition in the complex setting Analogous definition in C n A real subspace V of C n is protohomogeneous if there is a (connected) subgroup of U( n ) that acts transitively on the unit sphere of V Analogous problem in C n Classify protohomogeneous subspaces of C n { e 1 , . . . , e n } C -orthonormal basis of C n , J complex structure of C n Totally real subspaces V = span R { e 1 , . . . , e k } are protohomogeneous ❀ SO( k ) ⊂ U( n ) acts transitively on S k − 1

Getting intuition in the complex setting Analogous definition in C n A real subspace V of C n is protohomogeneous if there is a (connected) subgroup of U( n ) that acts transitively on the unit sphere of V Analogous problem in C n Classify protohomogeneous subspaces of C n { e 1 , . . . , e n } C -orthonormal basis of C n , J complex structure of C n Totally real subspaces V = span R { e 1 , . . . , e k } are protohomogeneous ❀ SO( k ) ⊂ U( n ) acts transitively on S k − 1 Complex subspaces V = span C { e 1 , . . . , e k } are protohomogeneous ❀ U( k ) ⊂ U( n ) acts transitively on S 2 k − 1

Getting intuition in the complex setting Analogous definition in C n A real subspace V of C n is protohomogeneous if there is a (connected) subgroup of U( n ) that acts transitively on the unit sphere of V Analogous problem in C n Classify protohomogeneous subspaces of C n { e 1 , . . . , e n } C -orthonormal basis of C n , J complex structure of C n Totally real subspaces V = span R { e 1 , . . . , e k } are protohomogeneous ❀ SO( k ) ⊂ U( n ) acts transitively on S k − 1 Complex subspaces V = span C { e 1 , . . . , e k } are protohomogeneous ❀ U( k ) ⊂ U( n ) acts transitively on S 2 k − 1 V = span R { e 1 , Je 1 , e 2 } is not protohomogeneous ❀ N U ( n ) ( V ) = U(1) × U( n − 2) does not act transitively on S 2

Getting intuition in the complex setting π : C n → V orthogonal projection, V real subspace of C n , v ∈ V \ { 0 } Definition The K¨ ahler angle of v with respect to V is the angle ϕ ∈ [0 , π/ 2] between Jv and V . Equivalently, � π Jv , π Jv � = cos 2 ϕ � v , v � . V has constant K¨ ahler angle ϕ if the K¨ ahler angle of any v ∈ V \ { 0 } with respect to V is ϕ .

Getting intuition in the complex setting π : C n → V orthogonal projection, V real subspace of C n , v ∈ V \ { 0 } Definition The K¨ ahler angle of v with respect to V is the angle ϕ ∈ [0 , π/ 2] between Jv and V . Equivalently, � π Jv , π Jv � = cos 2 ϕ � v , v � . V has constant K¨ ahler angle ϕ if the K¨ ahler angle of any v ∈ V \ { 0 } with respect to V is ϕ . Totally real subspaces have constant K¨ ahler angle π/ 2 Complex subspaces have constant K¨ ahler angle 0 V = span R { e 1 , Je 1 , e 2 } does not have constant K¨ ahler angle

Getting intuition in the complex setting π : C n → V orthogonal projection, V real subspace of C n , v ∈ V \ { 0 } Definition The K¨ ahler angle of v with respect to V is the angle ϕ ∈ [0 , π/ 2] between Jv and V . Equivalently, � π Jv , π Jv � = cos 2 ϕ � v , v � . V has constant K¨ ahler angle ϕ if the K¨ ahler angle of any v ∈ V \ { 0 } with respect to V is ϕ . Totally real subspaces have constant K¨ ahler angle π/ 2 Complex subspaces have constant K¨ ahler angle 0 V = span R { e 1 , Je 1 , e 2 } does not have constant K¨ ahler angle Proposition [Berndt, Br¨ uck (2001)] V ⊂ C n is protohomogeneous if and only if it has constant K¨ ahler angle.

Getting intuition in the complex setting π : C n → V orthogonal projection, V real subspace of C n , v ∈ V \ { 0 } Definition The K¨ ahler angle of v with respect to V is the angle ϕ ∈ [0 , π/ 2] between Jv and V . Equivalently, � π Jv , π Jv � = cos 2 ϕ � v , v � . V has constant K¨ ahler angle ϕ if the K¨ ahler angle of any v ∈ V \ { 0 } with respect to V is ϕ . Totally real subspaces have constant K¨ ahler angle π/ 2 Complex subspaces have constant K¨ ahler angle 0 V = span R { e 1 , Je 1 , e 2 } does not have constant K¨ ahler angle Proposition [Berndt, Br¨ uck (2001)] V ⊂ C n is protohomogeneous if and only if it has constant K¨ ahler angle. Moreover, V has constant K¨ ahler angle ϕ ∈ [0 , π/ 2) if and only if V = span { e 1 , cos ϕ Je 1 + sin ϕ Je 2 , . . . , e k , cos ϕ Je 2 k − 1 + sin ϕ Je 2 k } .

Back to the quaternionic setting Problem Classify protohomogeneous subspaces of H n .

Back to the quaternionic setting Problem Classify protohomogeneous subspaces of H n . J ⊂ End R ( H n ) quaternionic structure of H n { J 1 , J 2 , J 3 } canonical basis of J : J 2 i = − Id and J i J i +1 = J i +2 = − J i +1 J i