Manifold structures on Morse stable and unstable sets

For a globally C² function on a finite-dimensional real inner product space with globally Lipschitz gradient, the stable and unstable sets of a nondegenerate critical point carry C¹ manifold structures. Their topology is the subspace topology, and their inclusions into the ambient space are C¹. The model spaces are the positive and negative Hessian spectral subspaces, of dimensions finrank E - morseIndex f x and morseIndex f x, respectively.

The global straightening charts from IsNondegenerateCriticalPoint.exists_stableSet_chart and exists_unstableSet_chart supply the local normal forms. The linear-slice atlas construction assembles them without choosing a basis or changing the topology. These manifold structures allow stable and unstable sets to be used as domains of manifold maps, as required when studying their transverse intersections and connecting trajectories.

The results concern complete negative-gradient flows on a vector space; they do not assert the stable-manifold theorem for an arbitrary vector field or on an arbitrary manifold.

References

• M. Audin and M. Damian, Morse Theory and Floer Homology, Springer Universitext, 2014, Chapter 2.

theorem TauCeti.IsNondegenerateCriticalPoint.exists_stableSet_isManifold {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {f : E → ℝ} {x : E} {K : NNReal} (h : IsNondegenerateCriticalPoint f x) (hfs : ContDiff ℝ 2 f) (hf : LipschitzWith K (gradient f)) : ∃ (C : ChartedSpace ↥⋯.stableLinearSubspace ↑((negativeGradientFlow f hf).stableSet x)), IsManifold (modelWithCornersSelf ℝ ↥⋯.stableLinearSubspace) 1 ↑((negativeGradientFlow f hf).stableSet x) ∧ ContMDiff (modelWithCornersSelf ℝ ↥⋯.stableLinearSubspace) (modelWithCornersSelf ℝ E) 1 Subtype.val

The global stable set of a Morse critical point has a C¹ manifold structure modelled on the stable Hessian subspace, with its subspace topology and a C¹ inclusion into the ambient space. The dimension of the model plus the Morse index equals the ambient dimension.

theorem TauCeti.IsNondegenerateCriticalPoint.exists_unstableSet_isManifold {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] {f : E → ℝ} {x : E} {K : NNReal} (h : IsNondegenerateCriticalPoint f x) (hfs : ContDiff ℝ 2 f) (hf : LipschitzWith K (gradient f)) : ∃ (C : ChartedSpace ↥⋯.unstableLinearSubspace ↑((negativeGradientFlow f hf).unstableSet x)), IsManifold (modelWithCornersSelf ℝ ↥⋯.unstableLinearSubspace) 1 ↑((negativeGradientFlow f hf).unstableSet x) ∧ ContMDiff (modelWithCornersSelf ℝ ↥⋯.unstableLinearSubspace) (modelWithCornersSelf ℝ E) 1 Subtype.val

The global unstable set of a Morse critical point has a C¹ manifold structure modelled on the unstable Hessian subspace, with its subspace topology and a C¹ inclusion into the ambient space. The model dimension is the Morse index.