Lipschitz Geometry of Mixed Pham-Brieskorn Singularities

Technical Summary: Lipschitz Geometry of Mixed Pham-Brieskorn Singularities

Problem Statement
The paper investigates the Lipschitz geometry of mixed singularities of Pham-Brieskorn type. These are defined by function germs $f_{a,b}: (\mathbb{C}^n, 0) \to (\mathbb{C}, 0)$ of the form:
$$f_{a,b} = z_1^{a_1+b_1}\bar{z}_1^{b_1} + \dots + z_n^{a_n+b_n}\bar{z}_n^{b_n}$$
where $a = (a_1, \dots, a_n)$ and $b = (b_1, \dots, b_n)$ are integer vectors with $0 < a_1 \le \dots \le a_n$ and $b_i \ge 0$. The associated holomorphic Pham-Brieskorn polynomial is $g_a = z_1^{a_1} + \dots + z_n^{a_n}$.

While the topological relationship between $f_{a,b}$ and $g_a$ is well-established (they are topologically equivalent under certain conditions, and $f_{a,b}$ admits a Milnor fibration), the paper addresses the gap between topological and analytic classifications. Specifically, it seeks to determine conditions under which two mixed Pham-Brieskorn germs are bi-Lipschitz equivalent. The central question is whether the bi-Lipschitz type is determined solely by the exponent vectors $a$ and $b$, and whether topological equivalence implies bi-Lipschitz equivalence within this family.

Methodology
The author employs tools from Lipschitz geometry, singularity theory, and the theory of weighted homogeneous germs. The methodology proceeds through several key steps:

1. Topological Normal Forms: The paper utilizes a homeomorphism $\phi$ to relate $f_{a,b}$ to a "topological normal form." If $a_i \ge 1$ for all $i$, $f_{a,b}$ is topologically equivalent to $g_a$. If some $a_i=0$, it is equivalent to a mixed form involving real norms $\|z_i\|^{2b_i}$.
2. Multiplicity and Splitting: To analyze bi-Lipschitz equivalence, the author adapts a version of the "Splitting Lemma" for singularities. Theorem 4.1 establishes that for smooth function germs with separated variables and isolated singularities, if the multiplicities of the components are strictly ordered, the multiplicities of corresponding components in a bi-Lipschitz equivalent germ must match.
3. Tangent Cones and Invariants: For the case of mixed surfaces ($n=2$), the paper analyzes the tangent cones of the zero sets $X_{a,b} = f_{a,b}^{-1}(0)$. It introduces the invariant $\mu_{a,b} = \frac{a_2 + 2b_2}{a_1 + 2b_1}$ to classify the geometry of the tangent cone.
4. Arc Contact and Inner/Outer Metrics: The classification of inner bi-Lipschitz geometry relies on the contact order of semialgebraic arcs (using the $\beta$-horn classification by Birbrair). For outer geometry, the paper examines the Lipschitz normal embedding property and the behavior of paths connecting points on different branches of the singularity.

Key Contributions and Results

• Bi-Lipschitz Classification of Mixed Pham-Brieskorn Germs:
  The paper proves that the bi-Lipschitz type of $f_{a,b}$ is determined by the vectors $a$ and $b$.

  • Theorem 4.3: If $f_{a,b}$ and $f_{c,d}$ are bi-Lipschitz equivalent, then the exponent vectors must satisfy specific permutation relations. Specifically, if $a_i, c_i \ge 2$, then $a=c$ and the sums $a_i + 2b_i$ must match $c_{\sigma(i)} + 2d_{\sigma(i)}$ for some permutation $\sigma$.
  • Corollary 4.4: A mixed polynomial $f_{a,b}$ is bi-Lipschitz equivalent to its holomorphic counterpart $g_c$ if and only if $b=0$ (i.e., the mixed term vanishes), except in the case of topological submersions where specific linear relations hold.

• Topological Triviality vs. Distinct Lipschitz Types:
  A major contribution is the construction of infinite families of mixed singularities that are topologically trivial (equivalent to a standard form) but possess distinct bi-Lipschitz types.

  • Proposition 5.1: For a fixed vector $a$, there exist subfamilies of $f_{a,b}$ (varying $b$) that are all topologically equivalent to the same normal form but contain infinitely many distinct bi-Lipschitz classes. This demonstrates that Lipschitz geometry captures finer invariants than topology alone.

• Geometry of Mixed Surfaces ($n=2$):
  For surfaces defined by $f_{a,b}: (\mathbb{C}^2, 0) \to (\mathbb{C}, 0)$, the paper provides a detailed classification based on the invariant $\mu_{a,b}$:

  • Tangent Cones: The structure of the tangent cone $C(X_{a,b}, 0)$ is determined by $\mu_{a,b}$ and the values of $a_1, a_2$. For instance, if $\mu_{a,b} > 1$ and $a_1 \ge 1$, the tangent cone is the plane $\{z_1=0\}$.
  • Lipschitz Normal Embedding: The paper determines exactly when these surfaces are Lipschitz normally embedded (where the inner and outer metrics are equivalent).
    • Surfaces of "Type (2)" and "Type (5)" (homogeneous with isolated singularities) are normally embedded.
    • Surfaces of "Type (4)" with $a_2 \ge 2$ are not normally embedded.
    • Surfaces of "Type (1)" with $a_1 + 2b_1$ even are not normally embedded.
  • Outer Geometry Invariant: Theorem 6.8 establishes that for surfaces of Type (1), the value $\mu_{a,b}$ is an invariant of the subanalytic outer bi-Lipschitz geometry. If $\mu_{a,b} \neq \mu_{c,d}$, the surfaces are not outer bi-Lipschitz equivalent.

• Inner Geometry:
  The paper shows that most mixed surfaces are inner bi-Lipschitz equivalent to the standard metric cone (the $\beta$-horn with $\beta=1$). However, surfaces of Type (3) (where $a_1=0$ and $\mu_{a,b} > 1$) are inner equivalent to $\beta$-horns with $\beta = \mu_{a,b}$, providing a complete inner classification for this specific case.

Significance and Claims
The paper claims to provide a rigorous framework for distinguishing mixed Pham-Brieskorn singularities using Lipschitz invariants. It highlights that while these singularities often share the same topological type as their holomorphic counterparts (or simple real forms), their metric geometry is significantly richer and more sensitive to the specific exponents $a$ and $b$.

The significance lies in:

1. Refining Classification: Moving beyond topological equivalence to show that the "Lipschitz realm" lies strictly between topology and analytic geometry, offering a new set of invariants (like $\mu_{a,b}$ and the specific sums $a_i + 2b_i$).
2. Counterexamples to Intuition: Demonstrating that topological triviality does not imply bi-Lipschitz triviality, even within a highly structured family of polynomials.
3. Geometric Invariants: Identifying $\mu_{a,b}$ as a concrete, computable invariant for the outer geometry of mixed surfaces, which is determined solely by the exponents.

The work relies on established results from Milnor fibrations, tangent cone theory (Whitney, Seade, Ruas), and the classification of semialgebraic surfaces (Birbrair), extending them to the mixed function context. It does not propose new experimental applications but rather deepens the theoretical understanding of singularity geometry.