##  [Mean Curvature](/mean-curvature-1) 

 Definition

At a regular point of a smooth surface, the arithmetic mean H=(k1+k2)/2 of the two principal curvatures; an extrinsic measure of local bending related to the surface's normal variation.

 

 

 

 

 

 





## Principle

Principle

Mean curvature arises as the first variation of surface area: to first order, displacing a surface along its normal changes area proportionally to H, so H=0 characterizes stationary-area (minimal) surfaces.

 

 

 

 

 





## Demonstration

Demonstration

A round sphere of radius R has k1=k2=1/R so H=1/R; a minimal surface such as a catenoid or a soap film patch satisfies H=0 at every point.

 

 

 

 

## Misapplication

Misapplication

Using mean curvature as an intrinsic invariant or confusing H with Gaussian curvature; ignoring orientation sign which makes H change sign when the unit normal is reversed.

 

 

 

 

 





## Consequence

Consequence

Mean curvature governs physical equilibria of interfaces (soap films, capillarity) and drives geometric flows (mean curvature flow) that smooth shapes toward lower-area configurations.

 

 

 

 

## Reversal

Reversal

Replacing averaging by multiplication yields Gaussian curvature; switching to normal curvature along a direction extracts directional bending rather than the averaged normal variation.

 

 

 

 

 





## Boundary

Boundary

Defined for smooth regular surfaces with a chosen unit normal; undefined at singularities or nonorientable surfaces without a consistent normal. Sign conventions vary between authors.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension exists between extrinsic significance (depends on embedding and chosen normal) and its variational role (intrinsic functional derivative of area); one must distinguish geometric role from physical interpretation.

 

 

 

 

 





## Synthesis

Synthesis

Mean curvature is the average of the principal curvatures at a point, an extrinsic scalar measuring normal bending that dictates area variation, physical equilibrium shapes, and smoothing dynamics.