Variation of and minimal surfaces are key concepts in differential geometry. They explore how curves and surfaces can minimize length or area, respectively. These ideas connect to broader themes of optimization and variational problems in geometry.
Geodesics, which locally minimize arc length, and minimal surfaces, which locally minimize area, have wide-ranging applications. From physics to computer graphics, these mathematical objects help us understand natural phenomena and solve complex problems in various fields.
Definition of arc length
Arc length is a fundamental concept in differential geometry that measures the distance along a curve between two points
It is a way to quantify the length of a curved path in a metric space, such as a Riemannian manifold or a Euclidean space
Arc length is intrinsic to the curve itself and does not depend on the chosen to represent the curve
Arc length formula
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The arc length of a curve γ:[a,b]→Rn is given by the formula L(γ)=∫ab⟨γ′(t),γ′(t)⟩dt
This formula involves integrating the norm of the tangent vector γ′(t) with respect to the metric ⟨⋅,⋅⟩ along the curve
The arc length is invariant under reparametrization of the curve, meaning that different parametrizations of the same curve will yield the same arc length
For a curve in Euclidean space, the arc length formula simplifies to L(γ)=∫ab∑i=1n(dtdγi)2dt
Parametric curves
Parametric curves are a way to represent curves in a metric space using a function γ:[a,b]→M from an interval [a,b] to the space M
The parameter t∈[a,b] can be thought of as a "time" variable that traces out the curve as it varies from a to b
Parametric curves are useful for studying the geometry of curves, such as their length, curvature, and geodesic properties
Examples of parametric curves include circles (x(t)=cost,y(t)=sint), helices (x(t)=cost,y(t)=sint,z(t)=t), and Bézier curves used in computer graphics
Smooth curves
A curve γ:[a,b]→M is said to be smooth if its derivatives of all orders exist and are continuous
Smooth curves have well-defined tangent vectors, curvature, and other geometric properties at every point along the curve
The set of smooth curves forms an infinite-dimensional manifold, which is an important object of study in differential geometry
Smooth curves can be approximated by piecewise linear curves, which are used in numerical computations and discrete differential geometry
Variation of arc length
Variation of arc length is a technique used to find curves that minimize or maximize the arc length functional L(γ)=∫ab⟨γ′(t),γ′(t)⟩dt
It involves considering small perturbations or variations of a curve and analyzing how the arc length changes under these variations
The goal is to find curves that are critical points of the arc length functional, meaning that small perturbations do not change the arc length to first order
Variational problem
The problem of finding curves that minimize or maximize the arc length functional is an example of a variational problem
Variational problems involve finding functions or curves that optimize a given functional, such as the arc length, energy, or action functional
Variational problems arise in many areas of mathematics, physics, and engineering, such as classical mechanics, general relativity, and optimal control theory
Techniques from , such as the and the principle of least action, are used to solve variational problems
Euler-Lagrange equation
The Euler-Lagrange equation is a necessary condition for a curve γ to be a critical point of a functional J(γ)=∫abL(t,γ(t),γ′(t))dt
It states that the curve must satisfy the differential equation ∂γ∂L−dtd∂γ′∂L=0
For the arc length functional, the Euler-Lagrange equation leads to the geodesic equation, which characterizes curves that locally minimize arc length
The Euler-Lagrange equation is a powerful tool in variational calculus and has applications in classical mechanics (Hamilton's principle), general relativity (Einstein's equations), and quantum mechanics (Schrödinger's equation)
Geodesics
Geodesics are curves that locally minimize the arc length between two points in a metric space
They are the analogues of straight lines in Euclidean space and great circles on a sphere
Geodesics are characterized by the property that their acceleration vector is always perpendicular to their velocity vector, meaning that they have zero tangential acceleration
Examples of geodesics include straight lines in Euclidean space, great circles on a sphere, and curves of shortest distance on a surface or Riemannian manifold
Geodesic equations
The geodesic equations are a system of second-order differential equations that characterize geodesics in a metric space
For a curve γ(t)=(x1(t),…,xn(t)) in local coordinates, the geodesic equations are x¨k+Γijkx˙ix˙j=0, where Γijk are the Christoffel symbols of the metric
The Christoffel symbols encode information about the curvature of the space and how it affects the acceleration of geodesics
The geodesic equations can be derived from the Euler-Lagrange equation applied to the arc length functional or from the principle of parallel transport along the curve
Minimal surfaces
Minimal surfaces are surfaces that locally minimize the area functional A(S)=∫SdA among all surfaces with a given boundary curve
They are the two-dimensional analogues of geodesics and have important applications in geometry, physics, and engineering
Minimal surfaces have zero at every point, meaning that their average curvature in all directions is zero
Definition of minimal surface
A surface S in a Riemannian manifold M is called minimal if its mean curvature vanishes identically, i.e., H≡0 on S
Equivalently, a surface is minimal if and only if it is a critical point of the area functional under variations that fix the boundary of the surface
Minimal surfaces can be characterized as surfaces whose coordinate functions are harmonic functions, satisfying the Laplace equation Δxi=0
Examples of minimal surfaces include planes, helicoids, catenoids, and the gyroid surface found in certain butterfly wing scales
Mean curvature
The mean curvature H of a surface at a point is the average of the principal curvatures κ1 and κ2 at that point, i.e., H=21(κ1+κ2)
It measures how much the surface deviates from being flat in a given direction
The mean curvature is an extrinsic quantity that depends on the embedding of the surface in the ambient space
Surfaces with positive mean curvature (such as spheres) are called convex, while those with negative mean curvature (such as hyperboloids) are called saddle surfaces
Plateau's problem
is the problem of finding a with a given boundary curve, named after the Belgian physicist
It is a classic problem in geometric measure theory and calculus of variations, with important applications in soap films, capillary surfaces, and membrane physics
The existence and regularity of solutions to Plateau's problem have been extensively studied, leading to important results such as the Douglas-Rado theorem and the Federer-Fleming regularity theorem
Numerical methods, such as the finite element method and the level set method, have been developed to solve Plateau's problem computationally
Soap films and bubbles
Soap films and bubbles are physical realizations of minimal surfaces, formed by dipping a wire frame into a soap solution or blowing air into a soap film
They minimize the surface area for a given enclosed volume, subject to the constraint of the wire frame or the pressure difference across the film
The geometry of soap films and bubbles is governed by the Laplace-Young equation, which relates the mean curvature to the pressure difference and surface tension
Soap films and bubbles exhibit fascinating geometric properties, such as the double bubble conjecture (the shape that encloses two given volumes with the least total surface area) and the Weaire-Phelan structure (a complex periodic minimal surface used in foam physics)
Examples of minimal surfaces
There are many interesting examples of minimal surfaces that arise in geometry, physics, and nature
Some of the most well-known examples include the catenoid, helicoid, Enneper's surface, and Scherk's surface
These surfaces exhibit a rich variety of geometric and topological properties, such as self-intersection, periodicity, and symmetry
Catenoid
The catenoid is the minimal surface formed by rotating a catenary curve (the shape of a hanging chain) around an axis
It is the only minimal surface of revolution besides the plane, and has the parametrization x(u,v)=(coshucosv,coshusinv,u)
The catenoid has negative Gaussian curvature everywhere and is a ruled surface, meaning that it contains straight lines through every point
Catenoids appear in nature as the shapes of soap films spanning two circular rings, and have applications in architecture and engineering
Helicoid
The helicoid is the minimal surface formed by twisting a vertical strip of paper and joining its ends to form a spiral ramp
It has the parametrization x(u,v)=(sinhusinv,sinhucosv,u) and is the only ruled minimal surface besides the plane
The helicoid is a periodic surface with screw symmetry, meaning that it is invariant under a rotation and vertical translation
Helicoids appear in nature as the shapes of certain seaweed and fungi, and have applications in propeller and screw design
Enneper's surface
Enneper's surface is a self-intersecting minimal surface discovered by the German mathematician Alfred Enneper in 1864
It has the parametrization x(u,v)=(u−31u3+uv2,v−31v3+vu2,u2−v2) and is the conjugate surface of the catenoid
Enneper's surface has a unique triple point where three sheets of the surface intersect at a common point and tangent plane
It is an example of a complete regular minimal surface with finite total curvature and is embeddable in Euclidean space
Scherk's surface
Scherk's surface is a doubly periodic minimal surface discovered by the German mathematician Heinrich Scherk in 1834
It has the implicit equation ezcosx=cosy and can be parametrized using the Weierstrass ℘-function
Scherk's surface has a saddle shape with alternating positive and negative curvature, and contains two orthogonal families of straight lines
It is an example of a triply periodic minimal surface, meaning that it has translational symmetry in three independent directions, and arises in the study of crystal structures and foam geometry
Properties of minimal surfaces
Minimal surfaces have many interesting geometric and analytic properties that make them a rich subject of study in differential geometry
Some of the most important properties include the Gauss curvature, isothermal parameters, the Weierstrass-Enneper representation, and the existence of conjugate minimal surfaces
Gauss curvature
The Gauss curvature K of a surface at a point is the product of the principal curvatures κ1 and κ2 at that point, i.e., K=κ1κ2
It is an intrinsic quantity that measures how the surface curves in different directions and is related to the total curvature and topology of the surface by the Gauss-Bonnet theorem
For minimal surfaces, the Gauss curvature is always non-positive (K≤0), and vanishes identically (K≡0) only for planes
The Gauss curvature of a minimal surface satisfies the Gauss equation Δlogλ=−2Kλ2, where λ is the conformal factor of the metric
Isothermal parameters
Isothermal parameters are local coordinates (u,v) on a surface that make the metric conformal to the Euclidean metric, i.e., ds2=λ2(du2+dv2)
In isothermal parameters, the coordinate curves u=const and v=const intersect at right angles and have equal magnitude of the first fundamental form
Every minimal surface admits isothermal parameters, which can be found by solving the Beltrami equation Δx=2H∂n∂x for the coordinate functions x
Isothermal parameters are useful for studying the conformal geometry of minimal surfaces and for deriving the Weierstrass-Enneper representation formula
Weierstrass-Enneper representation
The Weierstrass-Enneper representation is a formula that generates all minimal surfaces in terms of two holomorphic functions ϕ(z) and ψ(z), called the Gauss map and the height differential
It states that every minimal surface can be parametrized as x(z)=Re∫z(ϕ(z),ϕ(z)2,ψ(z))dz, where z=u+iv is a complex parameter and Re denotes the real part
The Weierstrass-Enneper representation provides a powerful tool for constructing and classifying minimal surfaces, and has led to the discovery of many new examples and families of minimal surfaces
It also reveals the close connection between minimal surfaces and complex analysis, as many properties of minimal surfaces can be studied using techniques from complex function theory
Conjugate minimal surfaces
Every minimal surface S has a conjugate minimal surface S∗, which is obtained by rotating the principal directions of S by 90 degrees at each point
The conjugate surface has the same first fundamental form as the original surface, but the opposite second fundamental form, i.e., I∗=I and II∗=−II
The coordinate functions of S and S∗ are harmonic conjugates, satisfying the Cauchy-Riemann equations ∂u∂x∗=∂v∂x and ∂v∂x∗=−∂u∂x
Examples of conjugate minimal surfaces include the catenoid and Enneper's surface, the helicoid and the conjugate helicoid, and the Scherk surfaces of different periods
The existence of conjugate minimal surfaces is a manifestation of the integrable nature of the minimal surface equations and has applications in the study of constant mean curvature surfaces and harmonic maps
Generalizations and applications
The theory of minimal surfaces has been generalized in many directions, leading to the study of higher-dimensional minimal submanifolds, minimal surfaces in Riemannian manifolds, and other variational problems in geometry and physics
Minimal surfaces have also found numerous applications in fields such as architecture, materials science, and computer graphics
Higher-dimensional minimal submanifolds
A minimal submanifold is a submanifold of a Riemannian manifold that locally minimizes the volume functional among all submanifolds with the same boundary
Examples of minimal submanifolds include geodesics in Riemannian manifolds, minimal surfaces in Euclidean space, and special Lagrangian submanifolds in Calabi-Yau manifolds
The theory of minimal submanifolds is a rich and active area of research, with connections to geometric measure theory, partial differential equations, and calibrated geometry
Important results in the theory of minimal submanifolds include the monotonicity formula, the compactness theorem, and the regularity theory for area-minimizing currents
Minimal surfaces in Riemannian manifolds
The theory of minimal surfaces can be extended to surfaces in Riemannian manifolds, where the notion of area is defined using the intrinsic metric of the manifold
Examples of minimal surfaces in Riemannian manifolds include geodesics in surfaces, minimal spheres in spheres, and minimal tori in flat tori
The existence and regularity of minimal surfaces in Riemannian manifolds is a difficult problem that depends on the geometry and topology of the ambient manifold
Important results in this area include the Douglas-Morrey solution
Key Terms to Review (16)
Arc Length: Arc length is the measure of the distance along a curve between two points, and it's essential for understanding the geometry of curves in various contexts. The calculation of arc length involves integrating the speed of a parametrized curve over a given interval, which connects deeply to concepts like reparametrization and the analysis of curve properties. Additionally, arc length plays a key role in studying variations and minimal surfaces, as well as the properties of curves described by Frenet-Serret formulas.
Bernhard Riemann: Bernhard Riemann was a German mathematician whose work laid the foundation for Riemannian geometry and significantly advanced the study of differential geometry. His ideas are essential for understanding concepts like curvature, geodesics, and the mathematical properties of curved spaces, connecting various aspects of geometry to physics and other areas.
Calculus of Variations: Calculus of variations is a field of mathematical analysis that deals with finding the extrema (minimum or maximum) of functionals, which are mappings from a set of functions to real numbers. It plays a crucial role in determining optimal paths and surfaces by examining how small changes in functions affect their associated functionals. This concept is particularly important when studying properties of geodesics and minimal surfaces, as it provides the tools to analyze how these structures minimize length or area.
Differential area element: A differential area element is a mathematical concept that represents an infinitesimally small piece of area on a surface or in a region of space. It is crucial for calculating integrals over surfaces and for understanding how properties like curvature and surface area behave locally. This concept is particularly important in variations of arc length and minimal surfaces, where it helps quantify how small changes in surface shape affect overall properties.
Euler-Lagrange Equation: The Euler-Lagrange equation is a fundamental equation in the calculus of variations, used to find functions that minimize or maximize functionals. It connects the concepts of variation, derivatives, and optimality by providing a necessary condition for a function to be an extremal of a functional, like arc length or surface area.
Geodesic Curvature: Geodesic curvature measures how a curve deviates from being a geodesic on a surface. It quantifies the bending of a curve as it moves along a surface, indicating how much the curve 'turns' compared to the shortest path between two points. This concept is vital when analyzing the behavior of curves on surfaces and is closely linked to geodesic equations and the principles of minimizing arc length, especially in contexts involving minimal surfaces.
Jacobi Equation: The Jacobi equation is a second-order differential equation that describes the behavior of Jacobi fields along geodesics in a Riemannian manifold. It captures how nearby geodesics deviate from each other and provides insight into the geometric structure of the manifold, particularly in relation to stability and curvature. Understanding this equation is crucial for exploring variations in arc length and for analyzing the properties of minimal surfaces.
Joseph Plateau: Joseph Plateau was a Belgian physicist known for his pioneering work on the study of minimal surfaces, particularly through the creation of soap films. His experiments demonstrated how these surfaces minimize surface area for a given boundary, laying foundational principles for the understanding of minimal surfaces and their properties.
Mean Curvature: Mean curvature is a measure of the curvature of a surface at a point, defined as the average of the principal curvatures. It plays a crucial role in understanding the geometric properties of surfaces, including their shapes and stability, and is closely related to concepts like the first and second fundamental forms, Gaussian curvature, and minimal surfaces.
Minimal Surface: A minimal surface is a surface that locally minimizes area for a given boundary, characterized by having zero mean curvature at every point. These surfaces arise naturally in various contexts, particularly in the study of geometric properties of manifolds and variational problems, linking them closely to fundamental forms, induced metrics, and curvature concepts in differential geometry.
Parametrization: Parametrization refers to the process of expressing a mathematical object, such as a curve or surface, in terms of one or more variables, called parameters. This approach allows for a flexible representation of geometric objects and aids in understanding their properties by transforming complex shapes into manageable equations that describe them based on the chosen parameters.
Plateau's Problem: Plateau's Problem is a classic issue in the field of differential geometry that seeks to find a minimal surface that spans a given contour or boundary in space. This problem highlights the relationship between the geometry of surfaces and their physical properties, particularly in understanding how minimal surfaces behave under various constraints, leading to the study of concepts such as minimal area and the calculus of variations.
Stable Minimal Surface: A stable minimal surface is a surface that minimizes area among all nearby surfaces while also possessing a property called stability, which ensures that small perturbations do not lead to an increase in area. These surfaces are characterized by having non-negative second variation of area, meaning they can withstand slight changes without becoming unstable or increasing in area. This concept connects to the broader study of minimal surfaces and their variations in arc length.
Unstable minimal surface: An unstable minimal surface is a type of surface in differential geometry that minimizes area locally but is not stable under small perturbations. While it achieves minimal area, any slight deformation leads to an increase in area, indicating that it does not resist changes well, unlike stable minimal surfaces. This concept is crucial when studying variations of arc length and understanding the behavior of surfaces in a variational setting.
Variational Principle: The variational principle is a foundational concept in mathematics and physics that asserts that certain quantities are stationary (usually minimized or maximized) at optimal solutions, often leading to equations governing the behavior of systems. It connects various fields by providing a method to derive equations of motion, optimize shapes, and understand stability within the context of physical and geometric problems.
Weierstrass Representation Theorem: The Weierstrass Representation Theorem provides a powerful method for representing minimal surfaces in three-dimensional space using complex parameters. This theorem states that every minimal surface can be parametrized by a complex function, allowing the surface to be expressed in terms of a holomorphic function and its conjugate. The theorem not only highlights the connection between minimal surfaces and complex analysis but also emphasizes how variations in arc length influence the geometric properties of these surfaces.