Jupiter's Inner Moons

Closest to Jupiter (inward of the Galilean moons) are the four small moons:

Metis

Pronounced "MEE tis"
Metis was a Titaness who was the first wife of Zeus (Jupiter).
Discovered by Synnott in 1979 (Voyager 1).
Metis and Adrastea lie within Jupiter's main ring. They may be the source of the material comprising the ring.
Small satellites within a planet's rings are sometimes called "mooms".

Pronounced "a DRAS tee uh"
Adrastea, the distributor of rewards and punishments, was the daughter of Jupiter and Ananke.
Discovered by graduate student David Jewitt (working under Danielson) in 1979 (Voyager 1).
Metis and Adrastea orbit inside the synchronous orbit radius and inside the Roche limit. They may be small enough to avoid tidal disruption but their orbits will eventually decay.
Adrastea is one of the smallest moons in the solar system.

Pronounced "am al THEE uh".
Amalthea was the nymph who nursed the infant Jupiter with goat's milk.
Discovered by Barnard in 1892.
Amalthea and Himalia are Jupiter's fifth and sixth largest moons; they are about the same size but only 1/15 the size of next larger one, Europa.
Like most of Jupiter's moons, Amalthea rotates synchronously; its long axis is pointed toward Jupiter.
The reddish color of Amalthea's surface is apparently due to sulfur originating from Io.
Its size and irregular shape imply that Amalthea is a fairly strong, rigid body. Its composition is probably more like an asteroid's than like the Galilean moons.
Like Io, Amalthea radiates more heat than it receives from the Sun.

What role do these moons play in maintaining Jupiter's rings?
We know next to nothing about these moons. Even the basic mass and size data are not very accurate.
Where does Amalthea's extra heat come from? Tidal stress?

... Jupiter ... Metis / Adrastea / Amalthea / Thebe ... Io ...