Information Geometry and Complex Geometry: Gluing a Partial $\mathcal F$-completion in $n=2$

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Consider a family of functions

$$f_t(x) := e^{\frac{t}{\log x}}, \quad x \in (0,1), \quad t \in [1/2, 2],$$


each of which can be viewed as a smooth curve in the $(x, f)$-plane for fixed $t$. As $t$ varies, this defines a continuous family of curves, or equivalently, a surface embedded in the extended space $(x, f, t)$.


We now define a topological gluing operation by identifying the boundary curves corresponding to $t = 1/2$ and $t = 2$ in the $(x, f)$-plane. The identification is performed along straight lines of slope $+1$, i.e., lines of the form


$$f = x + b, \quad b \in [-1, 1].$$


Explicitly, we declare the point $(x, f_{1/2}(x))$ to be equivalent to $(x', f_2(x'))$ whenever both points lie on the same line of slope $+1$; that is, whenever


$$f_{1/2}(x) - x = f_2(x') - x',$$


so that both have the same intercept $b \in [-1, 1]$. This identification defines an equivalence relation that glues the endpoints of the parameter domain $t \in [1/2, 2]$ together along a $1$-parameter family of lines.


The result is a quotient topological space, denoted by $S$. Topologically, $S$ is homeomorphic to a $2$-sphere with two conical singularities, arising from the degenerate directions introduced by the gluing. This "pinched" geometry reflects the collapse of boundary curves along a finite set of matched directions in function space.


We next endow $S$ with a geometric structure. Each function $f_t(x)$ gives rise to a Mellin transform


$$F_t(s) := \mathcal{M}(f_t)(s) = \int_0^1 f_t(x) \, x^{s-1} \, dx,$$


defined for $s \in \mathbb{C}_+$. This evaluates to the closed-form expression


$$F_t(s) = 2\sqrt{\frac{t}{s}}\, K_1\big(2\sqrt{ts}\big),$$


where $K_1$ is the modified Bessel function of the second kind. The function $F_t(s)$ acts as a generating function for a Hermitian metric structure, such as


$$g_t(s) := |F_t(s)|^2 \, ds \otimes d\bar{s}, \quad \text{or} \quad g_t(s) := \partial_s \partial_{\bar{s}} \log |F_t(s)|^2.$$


Crucially, the family $F_t(s)$ satisfies the linear, scale-covariant partial differential equation


$$t^2 \frac{\partial^3 F}{\partial t^3} = s^2 \frac{\partial F}{\partial s},$$


which links third-order variation in the parameter $t$ to first-order variation in the spectral variable $s$. This equation defines a nontrivial flow on the function space, suggesting that the induced geometry on $S$ is governed by a PDE-driven foliation or flow. The equation is invariant under the dual scaling $t \mapsto \lambda t,\, s \mapsto \lambda^{-1} s$, indicating that the metric structure $g^{\sim}$ respects a form of scale duality reminiscent of Mellin symmetry or projective flatness.


Since the functions $f_t(x)$ have been glued along the boundary curves $t = 1/2$ and $t = 2$, the associated metrics $g_t(s)$ must also be identified accordingly. This yields a globally defined quotient metric $g^{\sim}$ on the surface $S$, defined by


$$g^{\sim}(s) := [g_t(s)] \quad \text{on } S,$$


inheriting its complex structure and regularity from the family $F_t(s)$. Thus, $S$ becomes not just a topological quotient of exponential functions, but a geometric surface equipped with a globally defined Hermitian structure, arising from a family of PDE-constrained Mellin-Bessel kernels.


A Question To Think About:


Given that the quotient surface $S$ inherits a Hermitian metric $g^{\sim}$ via the Mellin-transformed family $F_t(s)$, and that $F_t(s)$ satisfies the third-order PDE


$$t^2 \frac{\partial^3 F}{\partial t^3} = s^2 \frac{\partial F}{\partial s},$$


how does this differential equation descend to the quotient surface $S$ after gluing? Can we formulate a well-defined differential operator or flow on $S$ that reflects the original PDE structure, and how does this operator interact with the quotient metric $g^{\sim}$?

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