Geometric Morse theory ( and its complex analogy)

Question

In the literature are there some concept of geometric version of Morse or Picard Lefschets theory? That is the comparison of level sets as Riemannian submanifold not merely as topological manifolds.

In particular is there a complete classification of Polynomials $P(z,w): \mathbb{C}^2 \to \mathbb{C}$ such that all regular level sets are mutually isometric Riemannian manifolds when we consider them as $2$ dimensional submanifolds of $\mathbb{R}^4 \simeq \mathbb{C}^2$ where level sets inherit the standard metric $dx_1^2+dy_1^2+dx_2^2+dy_2^2$ of $\mathbb{R}^4$? In particular does $z^2+w^2$ satisfy this property? Here we identify $(z,w)\in \mathbb{C}^2$ with $(x_1,y_1,x_2,y_2) \in \mathbb{R}^4$ with $z=x_1+iy_1,\;\;w=x_2+iy_2$.

Answer 1

Here is an answer for the last question: The regular level set $L_c := \{(z,w) \in \mathbb{C}^2 : P(z,w) = c \}$ of $P(z,w)=z^2 + w^2$ is not isometric to the level set $L_{2c}$. To see this, notice that $L_c$ and $L_{2c}$ are complete Riemannian manifolds. The map $(z,w) \to (\sqrt{2}z,\sqrt{2}w)$ restricts to a homothety $\phi:L_c \to L_{2c}$ which is not an isometry. If $h: L_{2c} \to L_c$ is an isometry then $h \circ \phi$ is a homothetic transformation of $L_c$ which is not an isometry. Hence $L_c$ is locally Euclidean (see Lemma 2, page 242, Kobayashi-Nomizu Vol. I, 1963). Notice that the level sets $L_c$ are minimal submanifolds of $\mathbb{C}^2$. Finally, it is well-known (at least to the experts) that a minimal and Ricci flat submanifold of $\mathbb{R}^n$ is totally geodesic. Thus if $L_c$ and $L_{2c}$ are isometric then $z^2 + w^2 - c = 0$ is a complex line. This is a contradiction since the affine complex curve $L_c$ is regular and has degree 2 whilst complex lines had degree 1. Then $L_c$ and $L_{2c}$ are not isometric as Riemannian manifolds.