Unified Geometric Extremal Physics

Solution to the Plateau Problem under the Unified Geometric Extremal Physics Framework

— Physical Restatement, Paradigm Reduction, and Its Isomorphism with the Isoperimetric Problem and the Poincaré Conjecture

Author: Zhang Suhang, Luoyang

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Abstract (Concise)

By treating surface area as a potential energy functional and applying the principle of minimum energy, one can prove that the solution to the Plateau problem must be a surface of vanishing mean curvature (a minimal surface). This method follows exactly the same paradigm as the physical proof of the isoperimetric problem and Perelman's entropy-monotonicity proof of the Poincaré conjecture:

Define energy/entropy functional → Extremum principle → Constant/zero curvature → Unique geometric structure.

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I. Restatement of the Plateau Problem

Let Γ be a given simple closed curve in three-dimensional Euclidean space.

The Plateau problem:

Find the surface S bounded by Γ that has the minimum area among all such surfaces.

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II. Physical Restatement

Define the potential energy of a surface as its area:

E = \text{Area}(S)

The Plateau problem is equivalent to:

Minimize the energy E under the boundary constraint \partial S = \Gamma .

Comparison:

· Isoperimetric problem: maximize area ⇔ minimize E = -A 

· Plateau problem: minimize area ⇔ minimize E = A 

Both are energy minimization problems.

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III. Principle of Minimum Energy

According to the principle of minimum energy:

A stable equilibrium surface must satisfy

\delta E = 0

i.e., the first variation of energy vanishes.

Since E = \text{Area}(S) , this is equivalent to the vanishing of the first variation of area:

\delta \text{Area} = 0

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IV. Equilibrium Condition: Vanishing Mean Curvature

A classical result in surface calculus of variations:

The first variation of area vanishes if and only if the mean curvature H of the surface is identically zero:

H = 0

Physical interpretation:

This corresponds to the equilibrium state of surface tension on a soap film, where the net force at every point of the film is zero.

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V. Conclusion of the Plateau Problem

Under a fixed boundary Γ:

1. Energy = surface area

2. Minimization condition \delta E = 0 

3. Yields mean curvature H = 0 

4. Surfaces satisfying this condition are precisely minimal surfaces

Therefore, the solution to the Plateau problem is a minimal surface.

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Unified Logical Chain of the Three Problems

1. Isoperimetric Problem

E=-A \longrightarrow \min E \longrightarrow \kappa=\text{constant} \longrightarrow \text{Circle}

2. Plateau Problem

E=\text{Area} \longrightarrow \min E \longrightarrow H=0 \longrightarrow \text{Minimal surface}

3. Poincaré Conjecture (Perelman)

\text{Entropy } \mathcal{W} \longrightarrow \text{Monotonicity} \longrightarrow \text{Ricci soliton} \longrightarrow \mathbb{S}^3

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Final Conclusion

The isoperimetric problem, the Plateau problem, and the Poincaré conjecture are not three unrelated theories,

but rather three instances of the same universal physical–geometric paradigm across different dimensions:

Energy/entropy extremum → Curvature condition → Unique canonical geometry.